Let $\displaystyle \varphi : R \rightarrow [0,\inf) $ be a real valued function and assume that $\displaystyle \varphi $ is continuous at 0 with $\displaystyle \varphi(0) = 0$. Let $\displaystyle f : R \rightarrow R $ be a real valued function and assume that there exists a constant c >0 such that

$\displaystyle \mid f(x) - f(y) \mid \leq c\varphi(x-y) $

for all $\displaystyle x,y \in R$. Prove that f is continuous.