## Difference between Continuity and Derivatives.

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?

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 Recognitions: Gold Member Science Advisor Staff Emeritus 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.
 Recognitions: Gold Member 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.

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What "limit exists"? The limit you look at to determine if f(x) is continuous at x= a, is $\lim_{x\to a} f(x)$ while the limit you look at to determine if f(x) is differentiable at x= a is $\lim_{h\to a} (f(a+h)- f(a))/h$. 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.