Differentiability implies continuity
= Differentiability implies continuity
{title2=$f(x+h)-f(x)=h[f'(x)+o(1)]$}
If a real <function> has a finite <derivative> at $x$, its <difference quotient> tends to $f'(x)$ and is bounded for sufficiently small nonzero increments. Multiplying it by the increment shows $f(x+h)-f(x)\to0$, proving <continuity> at $x$. The converse fails: the <absolute value function> is continuous but not differentiable at zero.