Let $\phi:\mathbb{R}\to\mathbb{R}$ a continuous function. Fix $x_0\in\mathbb{R}$ and consider $$\psi:\mathbb{R}\to\mathbb{R},\ \psi(y)=\min_{\xi\in[x_0,y]}\phi(\xi)\ .$$ Is $\psi$ a continuous function? In particular does $\psi(y)\to\phi(x_0)$ as $y\to x_0$?

I think in general these questions has a negative answer: think to the function $$\phi(x)=\sin\frac{1}{x},\ \text{if } x\neq0 \quad,\quad \phi(0)=0\ $$ at point $x_0=0$.

But if we assume that $\phi$ has bounded variation on compact intervals, or that $\phi$ is $C^1$, can we hope to obtain a positive answer?

**Edit after richard's comment** The counter-example is not valid since the function is not continuous at $0$. So maybe is the hypothesis of continuity of $\phi$ sufficient?