If $f: \mathbb R \to \mathbb C$ satisfies $|f'(z)| \le M$, what can be said

Question

Let $f: \mathbb R \to \mathbb C$ be such that $|f'(z)| \le M$ for some constant $M$. what can be said about $f$?

this is not a homework problem. I am wondering if this can give something like lipschitz continuity.

thanks

Answer 1

If you write $f(t)=f_1(t)+i f_2(t)$, then \begin{align} |f(t)-f(s)|&=\left((f_1(t)-f_1(s))^2+(f_2(t)-f_2(s))^2\right)^{1/2}\\ \ \\ &=\left( f_1'(c_1)^2(t-s)^2+f_2'(c_2)^2(t-s)^2\right)^{1/2}\\ \ \\ &=\left( f_1'(c_1)^2+f_2'(c_2)^2\right)^{1/2}\,|t-s|\\ \ \\ &=(|f'(c_1)|+|f'(c_2)|)\,|t-s|\\ \ \\ &=2 M |t-s|. \end{align}