Differentiable Operator-Valued Function is Continuous

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December 2022:  It has been suggested that this page be renamed.
In particular: Suggestion to re-name page to "Differentiable Function With Values in Normed Space is Continuous" because I originally proved this result for operator-valued functions, but nothing about the proof changes if the co-domain is any normed space
To discuss this page in more detail, feel free to use the talk page.


Theorem

Let $\struct {X, \norm \cdot_X}$ be a normed vector space.

Let $f : I \to X$ be a map defined on an interval $I$.

Let $x_0 \in I$ such that $f$ is differentiable at $x_0$.


Then $f$ is continuous at $x_0$.


Proof

We have by hypothesis that the derivative $\map {f'} {x_0}$ of $f$ at $x_0$ exists.

Hence:

\(\ds \lim_{x \mathop \to x_0} \norm {\map f {x} - \map f {x_0} }_X\) \(=\) \(\ds \lim_{h \mathop \to 0} \norm {\map f {x_0 + h} - \map f {x_0} }_X\) re-write limit
\(\ds \) \(=\) \(\ds \lim_{h \mathop \to 0} \norm {h \cdot \dfrac {\map f {x_0 + h} - \map f {x_0} } h}_X\)
\(\ds \) \(=\) \(\ds \lim_{h \mathop \to 0} \size h \cdot \norm {\dfrac {\map f {x_0 + h} - \map f {x_0} } h}_X\) Norm Axiom $\text N 2$: Positive Homogeneity
\(\ds \) \(=\) \(\ds \lim_{h \mathop \to 0} \size h \cdot \norm {\dfrac {\map f {x_0 + h} - \map f {x_0} } h - \map {f'} {x_0} + \map {f'} {x_0} }_X\)
\(\ds \) \(\le\) \(\ds \lim_{h \mathop \to 0} \size h \paren {\norm {\dfrac {\map f {x_0 + h} - \map f {x_0} } h - \map {f'} {x_0} }_X + \norm {\map {f'} {x_0} }_X }\) Norm Axiom $\text N 3$: Triangle Inequality
\(\ds \) \(=\) \(\ds 0 \cdot \norm {\map {f'} {x_0} }_X = 0\) $f$ is differentiable at $x_0$ with derivative $\map {f'} {x_0}$ by assumption

Thus:

$\map f x \to \map f {x_0}$ as $x \to x_0$

The result follows by the definition of continuity in metric spaces and Metric Induced by Norm is Metric.

$\blacksquare$


Sources

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