The Derivative That Never Settles

As $f$ satisfies the conditions of the Mean-Value Theorem we know that for any interval $(a,b)$ there exists at least one point $c \in (a,b)$ such that

$f'(c) = \dfrac{f(b) - f(a)}{b - a}$ .

Now as we are given that $f'(c) \ne 0$ for any $c \in \mathbb{R}$ we can conclude that for any $a,b \in \mathbb{R}$ we have that $a \ne b$ implies that $f(b) - f(a) \ne 0 \Longrightarrow f(a) \ne f(b)$ , i.e., $f$ is one-to-one.

Since the function is differentiable over all real numbers, it is continuous.

Since the derivative of the function never passes through zero, it must be either always positive with $f'(x)>0$ or always negative with $f'(x)<0 .$ This means the function is monotone, and all monotone functions are one-to-one.