Differentiable Operator-Valued Function is Continuous

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
By hypothesis, the derivative $\map {f'} {x_0}$ of $f$ at $x_0$ exists.

We have:

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.