The hole exception is the only exception to the rule that continuity and limits go hand in hand, but it’s a huge exception. It’s also a bit odd to say that continuity and limits usually go hand in hand and to talk about this exception because the exception is the whole point. When you come right down to it, the exception is more important than the rule. Consider the two functions, r and s, shown here.
These functions have gaps at x= 2 and are obviously not continuous there, but they do have limits as x approaches 2. In each case, the limit equals the height of the hole.
The hole exception: The only way a function can have a regular, two-sided limit where it is not continuous is where the discontinuity is an infinitesimal hole in the function.
So both functions in the figure have the same limit as x approaches 2; the limit is 4, and the facts that r(2) = 1 and that s(2) is undefined are irrelevant. For both functions, as x zeros in on 2 from either side, the height of the function zeros in on the height of the hole — that’s the limit. This bears repeating:
The limit at a hole: The limit at a hole is the height of the hole.
“That’s great,” you may be thinking. “But why should I care?” Well, stick with this for just a minute. Suppose you drop a ball and you try to calculate its average speed during zero elapsed time. This would give you
is undefined, the result would be a hole in the function. Function holes often come about from the impossibility of dividing zero by zero. It’s these functions where the limit process is critical, and such functions are at the heart of the meaning of a derivative, and derivatives are at the heart of differential calculus.
The derivative-hole connection: A derivative always involves the undefined fraction
and always involves the limit of a function with a hole.