Differentiability Implies Continuity

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

Proof: Observe that $f(x)-f(a)=(x-a)[\frac{f(x)-f(a)}{x-a}]$ Take limits of both sides, $li{m}_{x\to a}[f(x)-f(a)]=0$

Related

• Continuous Function
• Differentiable Function