Definition:
A function f is continuous at a number x=a if
limx→af(x)=f(a).
Remember that limx→af(x) describes both
what is happening when x is slightly less than a and what is
happening when x is slightly greater than a. Thus there
are three conditions inherent
in this definition of continuity. A function is continuous at a if the limit as
x→a exists, and f(a) exists, and this limit is equal to
f(a). This means that the following three values are
equal: limx→a−f(x)=f(a)=limx→a+f(x)
I.e. the value as
x approaches a from the left is the same as the value as x
approaches a from the left (the limit exists) which is the same
as the value of f at a.
If any of these quantities is different, or if any of them fails to
exist, then we say that f(x) is discontinuous
at x=a, or that f(x) has a discontinuity
at x=a.
What does this mean graphically? If you trace f with a
pencil from left to right, as you approach x=a, you are at some
height L (because limx→a−f(x)=L. As
you go through the x-value a, your height is also L (because
f(a)=L). Now as you keep going with your pencil, you are
beginning this last stretch at height L (because
limx→a+f(x)=L).
DO: Sketch f(x)=√x
and let a=4. Find f(a),limx→a−f(x)=L, and limx→a+f(x). Now,
follow along your graph as stated in the previous paragraph,
looking at each condition as you go. Is f(x)=√x
continuous at x=a?
Now, more interestingly, consider x2−1x−1
in this video:
Definition:
A function f is
continuous from the right at x=a if limx→a+f(x)=f(a), and is continuous from the left at x=a if
limx→a−f(x)=f(a), and is
continuous on an intervalI if it is
continuous at each interior point of I, is continuous from
the right at the left endpoint (if I has one), and is
continuous from the left at the right endpoint (if I has
one).