What does differentiable mean for a function?

1 Answer
Calculus V.
Jul 25, 2014

geometrically, the function f is differentiable at a if it has a non-vertical tangent at the corresponding point on the graph, that is, at (a,f(a)). That means that the limit
lim_{x\to a} (f(x)-f(a))/(x-a) exists (i.e, is a finite number, which is the slope of this tangent line). When this limit exist, it is called derivative of f at a and denoted f'(a) or (df)/dx (a). So a point where the function is not differentiable is a point where this limit does not exist, that is, is either infinite (case of a vertical tangent), where the function is discontinuous, or where there are two different one-sided limits (a cusp, like for f(x)=|x| at 0). See definition of the derivative and derivative as a function.

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