Question

What does differentiable mean for a function?

Answer 1

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.