Let \((X,d_x)\) and \((Y,d_y)\) be metric spaces and let \(f:X\to Y\) be a continuous function on $X$.
The modulus of continuity $\omega_f:\mathbb R_+\to\mathbb R_+$ is continuous on the set positive real numbers $\mathbb R_+$, in particular, the limit of $\omega_f$ at $0$ from above is $$\lim_{\delta\searrow 0} \omega_f(\delta)=0.$$
Proofs: 1