Difference between Continuity and Derivatives.

cjaylee

Hey. I am quite confused by continuity and derivatives. Both are finding the limits of a particular function as x approaches a. Then why is it that a graph that is continuous cannot be differentiable? If it is continuous, it means that the limit exists and so, it should be differentiable right?

Hurkyl

If p is three, does that mean q has to be three as well?

The limits used in the definitions of continuity and differentiability of a function f are different limits.

daniel_i_l

For example, a function with a "point" (f(x)=|x| has a point at x=0) can be continuous but not differentiable since the derivative is different on either side of the point.

HallsofIvy

What "limit exists"? The limit you look at to determine if f(x) is continuous at x= a, is [itex]\lim_{x\to a} f(x)[/itex] while the limit you look at to determine if f(x) is differentiable at x= a is [itex]\lim_{h\to a} (f(a+h)- f(a))/h[/itex]. It is easy to show that if a function is differentiable at x= a, it must be continuous but the other way is not true.