# Linear and non-differentiable

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.

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

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.