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2.14 The definition of continuity - YouTube
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in this video I will present the
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definition of continuity actually I will
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do a bit more I will write two
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equivalent frequence definitions of
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continuity I will present some non
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regulars ideas that explain the
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intuition behind the concept and I will
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explain how we define continuity
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differently in different domains I will
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begin with hand-wavy idea when we say
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that a function is continuous we mean
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that we can sketch its graph in one go
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without lifting the pen from the paper
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this is of course not rigorous but it
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helps for example this graph can be a
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sketch without lifting the pen from the
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paper I want to transform this vague
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idea into a rigorous definition let's
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examine some examples of graphs that
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cannot be drawn in one go and see why
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this graph for example cannot be a
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sketch with a lift in paraffin paper
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because it has a hole notice that it
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does not matter whether the function is
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undefined at the point or it is defined
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with the wrong value so to speak either
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way I want to say this function is not
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continuous this example is different it
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has a jump so I cannot a sketch it with
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our lifting pen from paper either
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neither can i this one because it has a
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vertical asymptote and finally I cannot
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sketch this one with a lifting pen from
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paper either I can sketch the graph fine
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on either sine of X equals 0 but it
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could not get through 0 because the
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function oscillates widely I want to say
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that all these examples are not
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continuous many of these examples of bad
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functions have one thing in common they
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lack a limit this function does not have
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a limit as X approaches 0
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this function does not have a limit as X
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approaches zero and this function does
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not have a limit as X approaches one
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however this example is different the
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function in this graph has a limit as X
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approaches a but it has a hole because
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the function is undefined on the other
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hand this function has a limit as X
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approaches a and it is define at a
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however the limit and the value are
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different and thus it still has a hole
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let's put all these examples together
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this which is how I want to define
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continuity I will say that a function f
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is continuous at a when the limit of f
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of X as X approaches a is equal to the
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value of the function f of a all the
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examples I constructed where I could not
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catch the graph of the function with a
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lift in pen from paper fail this so this
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seems like a good definition to make
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this into a precise definition I have to
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introduce my variables I will say that
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ace of real number and I will require F
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to be a function defined at least on an
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interval centered at a so that I can
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study its limited a to be very explicit
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this definition means 3 6 1 the limit
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must exist and therefore be a number 2
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the function must be defined and 3 the
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limit must be equal to the value of the
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function notice that both the limit and
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the functions must be the same number so
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for example if they both do not exist
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then they do not satisfy the definition
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and there we have it
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this is the definition of continuity
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this was the simplest way to write the
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definition but there is an alternative
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equivalent way we can use the
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epsilon-delta definition of limit
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explicitly here is what happens if we do
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it first as a reminder this is the
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definition of the limit as X approaches
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a of f of X is L for a function to be
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continuous the limit must not be just
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any number L but specifically f f/a
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so let's simply change L into f f/a
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and now we can do a simplification in
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the definition of limit we must exclude
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the case x equals a because we don't
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even care whether the function is
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defined at a but when we want continuity
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the function must be defined at a and
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the limit must be f f/a so we no longer
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need to exclude x equals a this
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condition at the button when we remove
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the exclusion x equals a is entirely
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equivalent and there we have it this is
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the second definition of continuity
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which is entirely equivalent to the
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first one it may recall the
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epsilon-delta definition of continuity
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it may appear more complicated but only
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because the first definition hides the
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Epsilon's and deltas inside the limit in
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any case it is useful to have multiple
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equivalent definitions in different
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context either one or the other may come
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in handy this process has highlighted an
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idea that while not rigorous is
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extremely useful to keep in mind here is
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in few words the difference between
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merely asking for a limit and asking for
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continuity when we say that the limit of
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f of x equals L roughly we mean that if
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X is close to a but not a then F of X is
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close to L but when we say that F is
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continuous at a roughly we mean that if
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X is close to a including a the Neph of
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X is close to FFA that is the difference
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so far I have explained what it means
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for a function to be continuous at a
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point here it is again but most of the
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time we want functions to be continuous
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on a whole domain in the final part of
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this video I will explain what that
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means if we say that a function is
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continuous on an open interval which
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simply means that it is continuous at
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every point on the interval however if
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we say that the function is continuous
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on a closed interval it is a bit
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different look at this example I want to
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say that the function in this graph is
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continuous on the closed interval a B
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because they certainly kind of sketched
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the graph with a lift in pen from paper
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but I cannot say that the limit exists
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at the endpoint a according to the
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definition we are using because the
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function is not even defined to the left
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of
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so instead I will require the following
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at all the points in the interior of the
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interval I require the function to be
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continuous at the left end point a I
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only need the right side limit of f of X
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to be equal to F of a and similarly at
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the right endpoint B I only need the
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left side limit of f of X to be equal to
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f of B and that's it that is how we
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define being continuous an open interval
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and being continuous on a closed
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interval of course there are other types
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of domains but you can figure out the
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corresponding definitions to be fair
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there is a better way to define
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continues another main that works for
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all domains at once and then we don't
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have to break it in two cases in further
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math courses for example if you study
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multivariable calculus you will need
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that by the definition but for our
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purposes this is simple and this is
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enough and finally one more concept if
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we ever simply say that a function is
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continuous without specifying where we
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mean continuous on its domain careful
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though because things like the following
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may happen look at the function in this
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graph it is continuous if I continues we
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mean continuous on its domain but it is
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not continuous at one this may sound
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like a contradiction but it is not if
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you pay attention to the precise
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definitions that we have given to the
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concepts
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