I know that the derivative in the title does not exist for any x but I do not have a clue why. Could someone explain why this derivative DNE at 0 and I can figure out why it does not exist at x? Thanks!
2026-03-30 10:56:40.1774868200
Frechet derivative of the sup norm function on $C[0,1]$
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$0$ is too easy: no directional derivative of the sup norm function exists there. It's harder to show that the Frechet derivative doesn't exist at, for example, the function $f(t) = t$.