Differentiable Functions Are Continuous

In the setting of Multivariable Calculus: Notation, let $n$, $m$ and $U$ be as in Multivariable Calculus: Notation (ii), with continuity as in Multivariable Calculus: Notation (iii), and let $a\in U$.

(i) Continuity. If $f:U\to\mathbb{R}^{m}$ or $f:U\to\mathbb{R}$ is differentiable at $a$, then $f$ is continuous at $a$.

(ii) Increment bound. If $f:U\to\mathbb{R}^{m}$ is differentiable at $a$ with derivative matrix $A$, there is a real number $\delta>0$ such that every $h\in\mathbb{R}^{n}$ with $\lVert h\rVert<\delta$ satisfies $a+h\in U$ and

$$\lVert f(a+h)-f(a)\rVert\le\bigl(\lVert A\rVert+1\bigr)\lVert h\rVert.$$

(iii) Increment bound, real-valued. If $f:U\to\mathbb{R}$ is differentiable at $a$ with derivative vector $w$, there is a real number $\delta>0$ such that every $h\in\mathbb{R}^{n}$ with $\lVert h\rVert<\delta$ satisfies $a+h\in U$ and

$$|f(a+h)-f(a)|\le\bigl(\lVert w\rVert+1\bigr)\lVert h\rVert.$$