Analysis - Prove Continuity of $ f(x) = f(x), f²(x) = (f(x))², f³(x) = (f(x))³$

Question

I've got a function $f: \Bbb R \rightarrow \Bbb R$.

$f², f³, . . .$ are pointwise defined as $f²(x) = (f(x))², f³(x) = (f(x))³, . . . .$

I have to prove/disprove

(i) $f$ is continuous so $f²$ and $f³$ are continuous

my attempt:

$f²(x) = (f(x))² = f(x) * f(x) \Rightarrow f²$ is continuous since factors are continuous

$f³(x) = (f(x))³ = (f(x))² * f(x) \Rightarrow f³$ is continuous since factors are continuous

also I have no idea how to prove/disprove:

(ii) $f²$ is continuous so $f$ is continuous

(iii) $f³$ is continuous so $f$ is continuous

Answer 1

i) Your attempt is correct.

Hints

ii) Think about a piecewise constant function which isn't continuous but whose square is constant.

iii) Use the function $x \mapsto \sqrt[3]{x}$ to prove that the assertion is true.

(Moreover, why using the function $x \mapsto \sqrt[3]{x}$ can't work in ii)?)