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

Question

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.

Please teach me about this contradiction or my wrong. Thank you for comments.

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Added

The inf-convolution I'm saying is $$ f^{\varepsilon}(x)=\inf\left\{f(y)+\frac{|x-y|^{2}}{\varepsilon}\right\}. $$

Answer 1

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.