Does Liouville's Theorem work in real analysis?

Question

Liouville's Theorem states that every bounded entire function must be constant. Does it work in real analysis? Justify your answer! I asked it because Liouville's Theorem is proved by complex analysis.

Answer 1

Take $f(x)=\sin{x}$. clearly $|f| \leq 1$ is bounded and entire but is not constant

Answer 2

Actually it does work in real analysis. The question is only which condition replaces the "entire" because it is certainly not true for all real-valued functions (take $\sin(x)$ as Chandru states). However, if a real-valued function $f$ is harmonic which means that:

$$\frac{\partial^2f}{\partial x_1^2} +\frac{\partial^2f}{\partial x_2^2} +\cdots +\frac{\partial^2f}{\partial x_n^2} = 0$$

It actually has the Liouville Property, isn't that neat?

Answer 3

I will again restate the question as I see it:

True or False: Every bounded, everywhere-differentiable, real-valued function must be constant.

This is false. I'm tempted to give you a counter-example, but that would be against the fundamental principles of patient problem-solving. I will instead follow Qiaochu's lead and note that you should consider some analytic real functions. A simple counter-example exists (simple meaning an elementary function).