Convex function

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a differentiable, convex and bounded function in $\mathbb{R}$. Prove that $f$ is constant.

Solution



Since $f$ is convex the tangent will always lie below the graph of the function with exception the tangency point. That is:

$$f(x) - f(x_0) \geq (x-x_0) f'(x_0) \tag{*}$$

From $(*)$ we have that:

$$\lim_{x \rightarrow  +\infty}f(x)=+\infty, \;\; f'(x_0)>0 \;\; {\rm and} \;\; \lim_{x \rightarrow -\infty} f(x)=+\infty , \;\; f'(x_0)<0$$

However , the above contradicts the boundness of $f$. Hence $f'(x)=0, \; \forall x \in \mathbb{R}$ and the result follows.