# Why is the inf-convolution of lower semicontinuous functions continuous?

I'm confusing now about the continuity of inf-convolution.

I understand that the inf-convolution of lower semicontinuous functions is semiconcave and so it's locally Lipschitz continuous (in particular, continuous.) However, if one consider a jump function like $$f(x)=1\quad(\text{for x<0}),\quad=0\quad(\text{for x\ge0}),$$
then the inf-convolution of this function is clearly discontinuous at $x=0$. I don't know why this difference occurs.

The inf-convolution I'm saying is $$f^{\varepsilon}(x)=\inf\left\{f(y)+\frac{|x-y|^{2}}{\varepsilon}\right\}.$$
I am assuming $\epsilon >0$ here.
You have $f^\epsilon(x) = \begin{cases} 1,& x < -\sqrt{\epsilon} \\ {x^2 \over \epsilon}, &-\sqrt{\epsilon} \le x < 0 \\ 0, & 0 \le x \end{cases}$, which is continuous.