Derivative Operator on Continuously Differentiable Function Space with Supremum Norm is not Continuous

Theorem
Let $I = \closedint 0 1$ be a closed real interval.

Let $\map C I$ be the real-valued, continuous on $I$ function space.

Let $\map {C^1} I$ be the continuously differentiable function space.

Let $x \in \map {C^1} I$ be a continuoulsly differentiable real-valued function.

Let $D : \map {C^1} I \to \map C I$ be the derivative operator such that:


 * $\forall t \in \closedint 0 1 : \map {Dx} t := \map {x'} t$

Suppose $\map C I$ and $\map {C^1} I$ are equipped with the supremum norm.

Then $D$ is not continuous.

Proof
$D$ is continuous.

We have that the derivative operator is linear mapping.

By continuity of linear transformations:


 * $\exists M \in \R_{> 0} : \forall x \in \map {C^1} I : \norm {\map D x}_\infty \le M \norm x_\infty$

Suppose $x = t^n$ with $n \in \N$.

Then:


 * $\norm {x}_\infty = \norm {t^n}_\infty = 1$


 * $\norm {x'}_\infty = \norm {n t^{n-1}}_\infty = n$

Hence:

In other words:


 * $\forall n \in \N : n \le M$

But $M$ is finite.

This is a contradiction.

Hence, $D$ is not continuous.