How to conclude from a homogeneous function $f$ of degree $1$ differentiable at $0$ that all norms are never differentiable at $0$?

Question

Assume the fact that a homogeneous function $f$ of degree 1 (i.e. a map from $E$ to $F$ such that $f(tx) = tf(x)$ for all $t > 0$) that is differentiable at $0$ is necessarily linear.

How do we conclude from this that all norms are never differentiable at the origin?

Answer 1

To illustrate the problem, let's prove the claim. Let $f : E \to F$ be homogeneous of degree $1$ and differentiable at the origin. Hence there exists a bounded linear map $A : E \to F$ such that $$0 = \lim_{h\to 0} \frac{f(0+h) - f(0) - Ah}{\|h\|} = \lim_{h\to 0} \frac{f(h) - Ah}{\|h\|}$$

Note that $f(0) = 0$, this follows from $f(0) = f(t0) = tf(0), \forall t > 0$.

Fix $x \ne 0$. Taking $h = tx$ for $t > 0$ we get $$0 = \lim_{t\to 0^+} \frac{f(tx) - A(tx)}{\|tx\|} = \lim_{t\to 0^+} \frac{tf(x) - tA(x)}{t\|x\|} = \frac{f(x)-Ax}{\|x\|}$$

so $f(x) = Ax$. Therefore $f \equiv A$ so it is linear.

The norm $\|\cdot\|$ is homogeneous of degree $1$ but not linear so it cannot be differentiable at the origin.