differentiable
Calling a function differentiable at a point is a promise that, if you zoom in close enough on its graph there, the curve looks indistinguishable from a straight line. Smooth, with no corner, no vertical cliff, no sudden break — under enough magnification it flattens into a line of definite slope.
Formally, f is differentiable at a if the limit f'(a) = lim_{h -> 0} (f(a + h) - f(a)) / h exists as a finite real number. An equivalent and very useful reformulation is the linear-approximation form: there is a number m and an error term o(h) with f(a + h) = f(a) + m h + o(h), where o(h)/h -> 0; then m = f'(a). A function differentiable at every point of a set is called differentiable on that set.
Differentiability is strictly stronger than continuity. Every function differentiable at a is continuous at a, but the converse fails badly: f(x) = |x| is continuous everywhere yet not differentiable at 0, and Weierstrass built a function continuous on all of R but differentiable nowhere. So 'smooth-looking' must be earned by the limit, not assumed from a hand-drawn graph.
f(x) = x^2 sin(1/x) for x not 0 and f(0) = 0 is differentiable at 0 with f'(0) = 0, yet f' is not continuous at 0 — differentiable does not force the derivative to be continuous.
Existence of f' is one thing; continuity of f' is a separate, stronger demand.
At an endpoint or boundary one uses a one-sided difference quotient, so 'differentiable on [a, b]' means two-sided in the interior and one-sided at the endpoints.