construct a function with dense critical value

Question

I am learning derivative of a function from $\mathbb R$ to $\mathbb R$. If $f$ has zero derivative at $x$, we call $x$ is a critical point of $f$, and $f(x)$ is critical value.

• If critical points $C(f)$ is dense in $\mathbb R$, can we say $f$ is a constant?

• Can we construct $f$, which saisfies that critical value $f(C(f))$ is dense in $\mathbb R$?

Any advice is helpful. Thank you.

Answer 1

For the first question, the answer is `yes'. We can use the intermediate value property of $f'$ to conclude that the whole $\mathbb{R}$ is the set of critical points.