Differentiable function

A real function $f$ is differentiable at $a$ if the finite limitexists. Equivalently, $f(a+h)=f(a)+f^{\prime }(a)h+o(|h|)$: a linear approximation has error smaller than the displacement. This gives the tangent line and implies continuity at $a$.

The converse fails: $|x|$ is continuous at zero, but its left and right derivatives are $-1$ and $1$. In several variables, differentiability requires one linear map approximating all small increments, rather than merely the existence of separate partial derivatives.