Minimum over a ball of a continuous function is continuous

I want to show that if $f:\mathbb{R}^n\to\mathbb{R}$ is continuous, then the function$$f_t(x)=\min_{\|x-y\|\leq t}f(y)$$is continuous for any $t>0$. I am having a hard time since the minimum point can't really be traced, and its tricky since the ball itself is moving, so I have to consider which points are removed and which points aren't. Any hints?