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A linear is defined as $$ F (x_1+x_2)=F (x_1)+F (x_2).$$ I want to see is there a linear function but non-differentiable? If it is please offer an example.

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    $\begingroup$ This kind of example can only occur in infinite dimension spaces. $\endgroup$
    – Zanzi
    Apr 19, 2016 at 14:01
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    $\begingroup$ That's an additive function, not a linear function $\endgroup$
    – user258700
    Apr 19, 2016 at 14:06

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On polynomial space with the norm $N(P)=\int_0^1 P^2$. $P\mapsto P(1)$ is a linear form. But if it is continuous then let's says its norm is $C$ and we have with $P=X^n$, $$ 1\leq \frac{C}{2n+1}$$ for all $n$. Absurd.

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