(a) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function, and let $x \in \mathbb{R}$. Define what it means for $f$ to be continuous at $x$. Show that $f$ is continuous at $x$ if and only if $f\left(x_{n}\right) \rightarrow f(x)$ for every sequence $\left(x_{n}\right)$ with $x_{n} \rightarrow x$.

(b) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a non-constant polynomial. Show that its image $\{f(x): x \in \mathbb{R}\}$ is either the real line $\mathbb{R}$, the interval $[a, \infty)$ for some $a \in \mathbb{R}$, or the interval $(-\infty, a]$ for some $a \in \mathbb{R}$.

(c) Let $\alpha>1$, let $f:(0, \infty) \rightarrow \mathbb{R}$ be continuous, and assume that $f(x)=f\left(x^{\alpha}\right)$ holds for all $x>0$. Show that $f$ must be constant.

Is this also true when the condition that $f$ be continuous is dropped?