Let $M_K$be the set of all continuous functions $f$ in $C_{[a,b]}$ satisfying a Lipschitz condition, i.e., the set of all $f$ such that $$ \left| f(t_1)-f(t_2)\right| \leq K \left| t_1-t_2\right| $$ for all $t_1,t_2 \in [a,b]$, where $K$ is a fixed positive number. I would like to prove that $M_K$ is the closure of the set of all differentiable functions on $[a,b]$ such that $\left|f'(t)\right|\leq K$.

My Try: I have proved that $M_K$ is a closed set and know that any such differentiable function is in $M_K$. Now I need to prove that for any $f\in M_K$ there exists a sequence of such differentiable functions $\{g_n\}$ such that $g_n\rightarrow f$. Any help?

By $C_{[a,b]}$, I mean the set of all continuous functions on the interval $[a,b]$ with distance:

$$ d(f,g) = \max_{a\leq t\leq b} \left|f(t)-g(t)\right| $$


Hint: Extend $f$ to a $K$-Lipschitz function on $\mathbb{R}$. Then use the standard method to approximate $f$ with a sequence $f_n$ obtained by convoluting with a sequence of mollifiers $$f_n(x) = \int_{\mathbb{R}} f(x-y)\omega_n(y) = \int_{\mathbb{R}} f(y)\omega_n(x-y)$$

It is easy to see that $f_n$ are also $K$-Lipschitz, and since they are smooth, $\,|f_n'| \le K$.


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