I could not find a way to start with, let alone solution. Any help would be greatly appreciated.
Let $M_K$ be the set of all functions f in $C_{[a,b]}$ satisfying a Lipschitz condition i.e., $|f(t_1)-f(t_2)|\leq K|t_1-t_2|$ for all $t_1,t_2\in[a,b]$, where $K$ is a fixed positive number. Then $M=\cup_K M_K$ is not closed.
If $a = b$, then every $f\in C([a,b])$ is Lipschitz continuous (with Lipschitz constant $0$), so in that degenerate case, $M$ actually is closed (trivially). Therefore let's assume $a < b$.
$M$ is the set of functions satisfying any Lipschitz condition on $[a,b]$. In particular, $M$ contains all polynomials, since these are continuously differentiable on all of $\mathbb{R}$, and hence their derivatives are bounded on $[a,b]$.
There is a theorem by Weierstraß that every continuous function on a compact interval can be uniformly approximated by polynomials, hence $M$ is dense in $(C([a,b]),\lVert\,\cdot\,\rVert_\infty)$. Since there are continuous functions that are not Lipschitz continuous, e.g. $x\mapsto \sqrt{\lvert x-c\rvert}$ for $c\in (a,b)$, $M$ is not closed (because the only closed and dense subset is the entire space).
In fact, since $M_K$ is closed and has empty interior for every $K$, and $M = \bigcup\limits_{n=1}^\infty M_n$ is a countable union of such sets, $M$ is meagre (of the first category) in $C([a,b])$, and by Baire's theorem $C([a,b])\setminus M$ is non-meagre (of the second category) in $C([a,b])$. So topologically, the set of Lipschitz continuous functions on $[a,b]$ is a small subset of the set of all continuous functions.
If your functions are $C^1$ then $f' \colon \left[a,b\right] \to \mathbb R$ will be a continuous function on a closed interval, so will be bounded. If the derivative is bounded by $K$ then the function is has a Lipschitz constant of at least $K$. To show the space is not closed you have to find a sequence of functions $(f_n)_{n=1} ^\infty$ in $\bigcup_K M_K$ which is Cauchy but does not converge to something in $\bigcup_K M_K$. The easiest way to do this is for your sequence to converge to a non-Lipschitz function. Certainly if the limit has a jump discontinuity then it will not be Lipschitz. Can you find a sequence of differentiable functions which converge to a discontinuous function?
