Differentiability Implies Continuity

Theorem

If X⊆R and f:X→R is differentiable at a, then f is continuous at a.

First consider:

limx→x0(f(x0)−f(x))=limx→x0(f(x0)−f(x)x0−x(x0−x))(1)=limx→x0(f(x0)−f(x)x0−x)limx→x0(x0−x)=f′(x0)limx→x0(x0−x)=f′(x0)×0=0

which implies that

limx→x0f(x)=f(x0).

The condition of differentiability allows the limit to be split at (1) since it exists.