differentiability

Differentiability is just a fancy word for 'smooth enough to have a well-defined slope here'. Run your finger along a graph: if at some point the curve bends gently, with no sudden kink or break, your finger keeps moving in a single clear direction — the curve is differentiable there. If instead you hit a sharp corner and your finger has to jerk to a new direction, it isn't.

Formally, a function f is differentiable at a point a when the derivative f'(a) = lim h->0 (f(a+h) - f(a)) / h exists as a finite number. For that limit to exist, the difference quotient has to approach the same value whether h shrinks from the positive side or the negative side — the slope coming in from the left must match the slope coming in from the right. A function is differentiable on an interval if this holds at every point of the interval.

The key relationship to remember runs one way only: differentiable implies continuous, but not the reverse. If a function has a derivative at a point, it must be continuous there (no jumps or holes). But plenty of continuous functions fail to be differentiable: |x| has a corner at 0, the cube-root curve has a vertical tangent at 0, and a cusp pinches to a point — all unbroken, yet with no single slope. So smoothness is a stronger demand than mere continuity.