Fix a closed interval $I$. Prove that the non-lipschitz functions in $C(I)$ are dense in $C(I)$.
I don't even know where to begin! This is a question from a previous exam in a Real Analysis course I am taking. Does anyone have any idea how to prove it? Perhaps using some harmonic analysis?
Fix any non-Lipschitz continuous function $g$. If $f$ is continuous then there exist polynomials $p_n$ converging uniformly to $f-g$. Now $g+p_n$ are non-Lipschtz functions converging to $f$.