Linear functional $f$ is continuous at $x_0=0$ if and only if $f$ is continuous $\forall x\in X$?

functional-analysis continuity

Suppose $f$ is continuous at some $x_0\in X$ and let $x_n\to x^*$, for some $x^*\in X$. Then the sequence $x_n-x^*+x_0\to x_0$ and thus $\|f(x_n)-f(x^*)\|=\|f(x_n-x^*+x_0)-f(x_0)\|\to 0$. Hence $f$ is continuous at $x^*$ as well.


Consider

$$\lVert T(x-x_0)\rVert=\lVert T(x)-T(x_0)\rVert$$

So if $\lVert x-x_0\rVert<\delta$ we have that $\lVert T(x)-T(x_0)\rVert=\lVert T(x-x_0)\rVert<\epsilon$ by continuity at $0$.

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