# Differentiability Implies Continuity

Theorem: If $f(x)$ is differentiable at $a$, then $f(x)$ is continuous at $a$.

Proof: Observe that $f(x)−f(a)=(x−a)[x−af(x)−f(a) ]$ Take limits of both sides, $lim_{x→a}[f(x)−f(a)]=0$

Theorem: If $f(x)$ is differentiable at $a$, then $f(x)$ is continuous at $a$.

Proof: Observe that $f(x)−f(a)=(x−a)[x−af(x)−f(a) ]$ Take limits of both sides, $lim_{x→a}[f(x)−f(a)]=0$