# Clarification on Some Notation in a Lemma: Approximating LSC functions by Lipschitz continuous functions

I'm reading a book and it states the following lemma:

Let $$c \in \mathbb{R}, u: E \rightarrow [c, \infty]$$ not identically equal to $$\infty$$ and define for $$t > 0$$: $$u_t(x) = \inf \{u(y) + t d(x,y) | y \in E\}$$ where $$d$$ is the distance in the metric space $$X$$. Then $$\text{Lip}(u_t) \leq t, u_t \leq u$$ and $$u_t(x)$$ increases to $$u(x)$$ as $$t \rightarrow \infty$$ whenever $$x$$ is a lower semicontinuous point of $$u$$.

I'm just having trouble with the notation:

1. What is the $$\inf$$ being taken over? If it's all $$y \in E$$ doesn't second term become zero -- so that I would always find $$u_t(x) = \inf \{u(y) + t d(x,y) | y \in E\} = u(x)$$? I guess, generally, I'm having trouble understanding what this function is.

2. What is $$\text{Lip}(u_t)$$? This was just some instantaneous notation.

• okay #2 is just the lipschitz constant of $u_t$, how about #1? – yoshi Feb 13 at 17:45