Any function that is non-proximable and non-smooth?

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Does anyone have any example, where

a function that is both non-proximable and non-smooth (in particular, non-Lipschitz continuous gradient)?

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You do not deliver a definition of "proximable", but I suspect that the function $$ x \mapsto (f(x), g(x))$$ is neither proximable or smooth if $f$ is not proximable and $g$ is not smooth.