Proof of Differentiable Functions Are Continuous

Throughout, the arithmetic and order of $\mathbb{R}$ are used through The Structure of ℝ as a Field and an Ordered Ring (i), The Structure of ℝ as a Field and an Ordered Ring (ii), Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (v) and Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (ix), and norms in $\mathbb{R}^{n}$ through Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (ii), Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (v) and Standard Basis Vectors of ℝ^q: Coordinates, Orthonormality and Unit Length (iii). We prove the two increment bounds first and deduce continuity from them.

Proof of (ii). Let $f:U\to\mathbb{R}^{m}$ be differentiable at $a$ with derivative matrix $A$. Applying Differentiability at a Point for Functions Between Euclidean Spaces (i) with tolerance $1$ gives a real $\delta>0$ such that every $h\in\mathbb{R}^{n}$ with $\lVert h\rVert<\delta$ satisfies $a+h\in U$ and $\lVert f(a+h)-f(a)-Ah\rVert\le\lVert h\rVert$. For such $h$ write

$$f(a+h)-f(a)=\bigl(f(a+h)-f(a)-Ah\bigr)+Ah .$$

By the triangle inequality, Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (vi), and the operator-norm bound $\lVert Ah\rVert\le\lVert A\rVert\,\lVert h\rVert$ of Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (ii),

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

Proof of (iii). The argument is the same. Let $f:U\to\mathbb{R}$ be differentiable at $a$ with derivative vector $w$; Differentiability at a Point for Functions Between Euclidean Spaces (ii) with tolerance $1$ gives a real $\delta>0$ such that every $h$ with $\lVert h\rVert<\delta$ satisfies $a+h\in U$ and $|f(a+h)-f(a)-w\cdot h|\le\lVert h\rVert$. Writing $f(a+h)-f(a)=\bigl(f(a+h)-f(a)-w\cdot h\bigr)+w\cdot h$ and bounding $|w\cdot h|\le\lVert w\rVert\,\lVert h\rVert$ by the Cauchy--Schwarz inequality, The Cauchy-Schwarz Inequality on $\mathbb{R}^n$ (i), we get, by the triangle inequality for $|\cdot|$ (Rules of Arithmetic and Order in an Ordered Field (xi)),

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

Proof of (i). Let $f$ be differentiable at $a$. By Differentiability at a Point for Functions Between Euclidean Spaces (iii), $f$ has a derivative matrix $A$ at $a$ if $f:U\to\mathbb{R}^{m}$, and a derivative vector $w$ at $a$ if $f:U\to\mathbb{R}$. Put $C=\lVert A\rVert+1$, respectively $C=\lVert w\rVert+1$, a positive real number, and let $\delta>0$ be given by (ii), respectively (iii); neither depends on a tolerance. Now let $\varepsilon>0$ be given, and only then put $\delta'=\min\{\delta,\varepsilon/C\}>0$. Let $x\in U$ with $d_{E}(a,x)<\delta'$ and put $h=x-a$, so that $x=a+h$ and $\lVert h\rVert=d_{E}(a,x)<\delta'\le\delta$. The increment bound then gives

$$d_{E}\bigl(f(x),f(a)\bigr)=\lVert f(a+h)-f(a)\rVert\le C\lVert h\rVert<C\cdot\frac{\varepsilon}{C}=\varepsilon ,$$

with $|f(x)-f(a)|$, the metric $d_{\mathbb{R}}$ of The Absolute Value Metric on the Real Numbers (i), in place of $d_{E}(f(x),f(a))$ when $f$ is real-valued. This is continuity of $f$ at $a$ in the sense of Continuous Map Between Metric Spaces (i).