I am reading the book "Convex Analysis" by R.T. Rockafellar. In page.117, it said that "the Lipschitz condition is satisfied by $f$ if and only if it is satisfied by $\text{cl}~f$".
I am not sure that I could understand this sentence exactly. First, if $f$ is Lipschitz continuous, it is obviously continuous and then we have $f = \text{cl}~f$. In this situation, $f$ has the same Lipschitz condition with $\text{cl}~f$.
However, if $\text{cl}~f$ is Lipschitz continuous, I think $f$ may not be Lipschitz continuous? For example, let \begin{equation} f(x) = \left\{ \begin{array}{ll} e^{-x}, \quad \text{if}~x > 0, \\ 10, \quad \text{if}~x = 0, \\ +\infty, \quad \text{otherwise}. \end{array} \right. \end{equation} Then its closed function $\text{cl}~f$ is \begin{equation} (\text{cl}~f)(x) = \left\{ \begin{array}{ll} e^{-x}, \quad \text{if}~x \ge 0, \\ +\infty, \quad \text{otherwise}. \end{array} \right. \end{equation} So the $\text{cl}~f$ is Lipschitz continuous but $f$ is not.
Here the definition of $\text{cl}~f$ is the closure of function $f$, i.e., the largest lower semi-continuous function majorized by $f$.
By the way, maybe there are some additional conditions that $f$ is proper and convex.