Exist $f\colon\mathbb R\longrightarrow\mathbb R$ is a monotonic, non-constant and periodic function of period $T>0$?

Question

Question

Say whether it is true or false that exist $f\colon\mathbb R\longrightarrow\mathbb R$ is a monotonic, non-constant and periodic function of period $T>0$ ?

Answer 1

Suppose $f:\mathbb R\to\mathbb R$ is increasing and has period $T$. Take any $x\in\mathbb R$, then $f(x) = f(x+T)$ implies $f$ is constant on $[x,x+T]$ due to monotonicity. So $f$ must be constant. Hence,

if $f\colon\mathbb R\longrightarrow\mathbb R$ is a monotonic, non-constant, periodic function of period $T>0$, then $f$ is continuous

is vacuously true.

Answer 2

a monotonic, non-constant, periodic function of period $T>0$, then $f$ is continuous.

It's a vacuous statement:

If $f$ is non-constant, then there are numbers $x$ and $y$ such that $$y>x \quad\text{ and } \quad f(y)\neq f(x)$$

Now $f$ is periodic, i.e. there exists $x'>y$ such that $f(x')=f(x)$ and hence:

• If $f(y) > f(x) = f(x')$ then $f$ is not monotonic.

• If $f(y) < f(x) = f(x')$ then $f$ is not monotonic, either.

Thus no such function exists, thus every such function is constant :-)