Your "proof" could be used for the converse: When $|f'(x)|\leq M$ for all $x\in[a,b]$ then $f$ is Lipschitz continuous on $[a,b]$, and $M$ can serve as a Lipschitz constant.
In fact we are asked to prove that there is an $M>0$ with $|f'(x)|\leq M$ for all $x\in[a,b]$, given that $f$ is Lipschitz continuous on $[a,b]$.
Now when $|f(y)-f(x)|\leq M\>|y-x|$ for all $x$, $y\in[a,b]$ then
$$\left|{f(y)-f(x)\over y-x}\right|\leq M\qquad(y\ne x)\ .$$
It follows that for each fixed $x\in[a,b]$ we have
$$\left|\lim_{y\to x}{f(y)-f(x)\over y-x}\right|\leq M$$
and therefore $|f'(x)|\leq M$.
The answer is that yes, such a function exists.
The starting point is the Pompeiu Derivative.. To summarize Wikipedia, Pompeiu constructed an everywhere differentiable strictly increasing function $g:[0,1]\rightarrow \mathbb{R}$ whose derivative $g'(x)$ is $0$ on a dense subset of $[0,1]$. Call this dense set $D$.
Now, let $h:\mathbb{R}\rightarrow (0,1)$ be your favorite diffeomorphism (e.g., you could pick $h(x) = \frac{1}{\pi}\arctan(x) + \frac{1}{2}$.)
Set $f(x) = g(h(x)) + x$. I claim that $f$ fulfills your criterion. By the chain rule, $f'(x) = g'(h(x))h'(x) + 1$. For $x\in h^{-1}(D)$, $g'(h(x)) = 0$, so $f'(x) = 1$ for $x\in h^{-1}(D)$. Note that since $h$ is a diffeomorphism, it is a homeomorphism, so $h^{-1}(D)\subseteq \mathbb{R}$ is dense. Lastly, since $g$ is strictly increasing, $f(x) -x$ is strictly increasing, so is not constant.
Best Answer
if a function $f$ is differentiable on an interval and the derivative is always different from zero then the function is either increasing or decreasing (the derivative has constant sign, by the Darboux property of derivatives).
Suppose a function $f$ has second derivative always different from zero. Apply 1. to the first derivative to find that the given function is either strictly concave or strictly convex. Take a point where the derivative is not zero, the tangent line is unbounded by above and below. So the function is unbounded because it lies either above or below the tangent line.