In Kolmogorov & Fomin's real analysis book specifically the section on convergence, open and closed sets in metric spaces, there is the following problem, which part a) asks to prove
The set of all functions on $[a,b]$ satisfying a Lipschitz condition of order $1$ (for some fixed constant $K$), denoted $M_K$ is 1) closed and 2) coincides with the closure of the set, denoted $D_K$, of all continuously differentiable functions on $[a,b]$ with first derivative bounded in absolute value by $K$ on all of $[a,b]$. (We are working in the space $C_{[a,b]}$ with metric $\max_{t\in [a,b]} |x(t)-y(t)|$)
After success with 1), I got stuck on 2) after proving that $cl(D_K)\subset M_K$, I failed to obtain the other set inclusion. After looking online for some suggestions, I noticed a lot of solutions used mollifiers and convolutions and sometimes mentioned a handful of higher spaces.
From what I recall, it's a matter of extending a given $K$-Lipschitz function $f$ to all the real line, so that it is $K$-Lipschitz on all of $\mathbb{R}$. Then you approximate the extension with a sequence obtained by convolutions with a mollifier and the extension. I understand that a mollifier is a smooth function that has compact support (identically zero outside a compact domain), has an integral over the whole real line equal to $1$ and satisfies a certain limit property with the dirac delta function, but I am not very familiar with the theory of distributions or generalized functions. I understand how to proceed with the extension of the original function but it is not apparent to me on how to show the convergence, and that the smoothness of the mollifier alone implies the smoothness of the sequence obtained by convolutions and hence satisfies the bounded condition on the first derivative. Am just I missing a lot of theory here?
Could someone explain these points or direct me towards some references so that I can learn them myself?
A function $f$ on $[a,b]$ is Lipschitz continuous with constant $M$ iff there is a constant $A$ and $g \in L^{\infty}[a,b]$ with $\|g\|_{\infty} \le M$ such that $$ f(x) = A+\int_{0}^{x}g(t)dt. $$ Can you show that $\{ f \in C[a,b] : |f| \le M \}\subset L^1$ is dense in $\{ f \in L^{\infty}[a,b] : \|f\|_{\infty} \le M \}\subset L^1$ in the $L^1$ norm? If you can, then you should be able to show what you want.