differentiable

A function has a well-defined derivative at a point — its graph has a single, unambiguous tangent line there.

$f$ is differentiable at $x=a$ when $f'(a)$ exists — the difference-quotient limit settles on a single value. Differentiability at a point requires continuity there, but continuity alone is not enough (a sharp corner or vertical tangent can still fail to be differentiable).

Examples

$f(x)=|x|$ is continuous at $x=0$ but not differentiable there — the left and right slopes disagree.