# Whether the $\lim _{x\to0}f(x)$ exists or not for $f(x)$ satisfying some properties.

Let $\ f:(0,1) \rightarrow \mathbb R$ be continuous function $s.t.$ $|f(x)-f(y)| \leq \lvert\cos(x)-\cos(y)\rvert$ for all $\ x,y \in (0,1)$ . Then

is it true that $\lim _{x\to0} f(x)$ exists .

MY APPROACH:

I supposed that $f(x)=[x], x \in (0,1)$. Here $[x]$ denotes the greatest integer not greater than x. This function satisfies all the criterions but the $\lim _{x\to0} f(x)$ does not exist. Am I correct ?

Please check my approach it is correct or not?

Even by your approach, $\lim_{x \to 0} f(x)$ exists and is equal to $0$. Because $f$ is defined only on $(0,1)$. If it was the case that the domain of $f$ was $(a,1)$ with $a \lt 0$ or $\Bbb R$, or something that takes an arbitrary small interval around the point $0$, then what you said would make sense.
Observe that $\cos$ satisfies Lipschitz condition and use that to show that $f$ is uniformly continuous on $(0,1)$. Now use extension theorem that if a function is uniformly continuous on $(a,b)$, then it is continuous on $[a,b]$ which will give you the answer about why the limit exists at the end point.