# Differentiable $f\colon I\to\mathbb{C}$ with bounded derivative is Lipschitz continuous

Prove that a differentiable function $f\colon I\to\mathbb{C}$ on an interval $I$ with bounded derivative is Lipschitz continuous, i.e. If $\lvert f'\rvert\leq L$ for some $L\in\mathbb{R}$, then for any $x_1,x_2\in I$ we have $$\lvert f(x_1)-f(x_2)\rvert\leqslant L\lvert x_1-x_2\rvert.$$

Despite two little things, I think the proof should work as follows:

I think, we can choose some $c\in\mathbb{C}$ with $\lvert c\rvert =1$ such that $$\lvert f(x_2)-f(x_1)\rvert = c\cdot (f(x_2)-f(x_1))~~(*)$$ since for $$\frac{f(x_2)-f(x_1)}{\lvert f(x_2)-f(x_1)\rvert}=:v$$ we have $\lvert v\rvert =1$ and then we can define $c:=v^{-1}$.

Next, in order to have equation $(*)$, the LHS has to be the real part of the RHS, i.e. $$\lvert f(x_2)-f(x_1)\rvert =\Re(c\cdot (f(x_2)-f(x_1)))=\varphi(x_2)-\varphi(x_1),$$ where $$\varphi:=\Re(cf).$$

Now, since $f$ is differentiable on $I$, it is, in particular, continuous on $[x_1,x_2]$, hence $\varphi$ is also continuous on $[x_1,x_2]$.

I am not sure about the following question:

Do we also have that $\varphi$ is differentiable on $(x_1,x_2)$? (Q)

Assuming that we can answer question (Q) with YES, we could apply the mean value Theorem on $\varphi$, telling us that $$\varphi(x_2)-\varphi(x_1)=(x_2-x_1)\varphi'(\xi)$$ for some $\xi\in (x_1,x_2)$.

By assumption, $\varphi'(\xi)=\Re(cf'(\xi))\leq\lvert cf'(\xi)\rvert\leq L$ and hence $$\lvert f(x_2)-f(x_1)\rvert = \varphi(x_2)-\varphi(x_1)=(x_2-x_1)\varphi'(\xi)\leqslant L\lvert x_2-x_1\rvert.$$

I am also not completely sure if $$\varphi' = \Re(cf')$$ is correct.

Despite the two things in the two yellow boxes, I am pretty sure the proof should work. It would be nice if you could give me some hints.

• You need to complex conjugate $c$. $\varphi$ is in general not differentiable if $0\in f((x_1, x_2))$. You could consider the squared absolute value. Commented Jan 6, 2017 at 21:32
• You mean $\lvert f(x_2)-f(x_1)\rvert = \bar{v} (f(x_2)-f(x_1))$?
– Rhjg
Commented Jan 6, 2017 at 21:36
• yes, or to circumvent the differentiability issue just set $c = \overline{f(x_2) - f(x_1)}$. Commented Jan 6, 2017 at 21:38
• Here is a related answer that does something similar math.stackexchange.com/a/2078872/27978. Essentially the idea is that if $\operatorname{re} (cz) \le K |c|$ for all $c$, then $|z| \le K$. In the above case, $K=L|x_1-x_2|$, $|c|=1$ and $z=f(x_1)-f(x_2)$. The answer to both of your yellow boxes is yes. Commented Jan 6, 2017 at 21:48
• @user251257 Sorry, but did not get why $\varphi$ isn't differentiable if $0\in f((x_1,x_2))$. Could you explain that, please?
– Rhjg
Commented Jan 6, 2017 at 22:20

Since $\operatorname{re}$ is continuous we have $\lim_{h \to 0} \operatorname{re} {cf(z+h)-cf(x) \over h} = \lim_{h \to 0} {\operatorname{re}(cf(z+h))-\operatorname{re}(cf(x)) \over h} = \operatorname{re} (cf'(x))$. That is, if $\phi(x) = \operatorname{re}(cf(x))$, then $\phi'(x) = \operatorname{re} (cf'(x))$.
• If $b$ is real, then $\operatorname{re} ({a \over b}) = {\operatorname{re} a \over b}$. Commented Jan 6, 2017 at 23:31