Weierstrass theorems on continuous functions on a closed interval

Give an example of a function $f: X \rightarrow \mathbb {R}$, continuous but not bounded on some ball, where $X$ is a complete metric space.

Let $M$ be a set in a complete metric space $X$ such that every real-valued function continuous on $M$ is bounded. Prove that $M$ is a compact.

Let $M$ be a set in a complete metric space $X$ such that every real-valued function continuous and bounded on $M$ attains its supremum and infimum. Prove that $M$ is a compact.

Prove that if the function $f(x)$ is continuous on the interval $a\leqslant x<+\infty$, and $\lim \limits_{x\to +\infty }f(x)$ is finite, then this function is bounded on the given interval.

Give an example of a function continuous on an interval:

Prove that if a function is defined and continuous on a closed interval, then its range is a closed interval.

The function $f$ is continuous on the closed interval $[a ; b]$. Prove that:

The function $f$ is continuous on the closed interval $[a ; b]$. Prove that the functions $m(x)$ and $M(x)$ are continuous on $[a ; b]$, if:

The function $f$ is continuous on $[a ;+\infty )$, and $\lim_{x \rightarrow +\infty } f(x)$ exists and is finite. Prove that $f$ is bounded on $[a ;+\infty )$.

The function $f$ is continuous on the interval $(a ; b)$ (finite or infinite), and $\lim_{x \rightarrow a+0} f(x)$ and $\lim_{x \rightarrow b-0} f(x)$ exist and are finite. Prove that the function $f$ is bounded on $(a ; b)$.

Prove the stated assertion. Every continuous function $f: X \rightarrow R$ defined on a compact set $X$ is bounded on it and attains its greatest and least values.

Prove Theorem 2.3.18.

Theorem 2.3.18: Let $(M,d)$ be a compact metric space and let $f: M \rightarrow \mathbb {R}$ be continuous. Then $f$ is bounded and it attains its extreme values on $M$.

Let $f: \mathbb {R} \rightarrow \mathbb {R}$, let $x_{0} \in \mathbb {R}$, and suppose that for every interval $(a, b)$ that contains $x_{0}$, the function $f$ is unbounded on ( $a, b$ ). Prove that $f$ is not continuous at $x_{0}$.

Let $f:[0, \infty ) \rightarrow \mathbb {R}$ be a continuous function such that $\lim_{x \rightarrow \infty } f(x)= L$, and $L<f(0)$. Prove that $f$ attains a maximum value in $[0, \infty )$.

Show that a non-empty set $E$ of real numbers is closed and bounded if and only if every continuous real-valued function on $E$ takes a maximum value.

Let the function $f(z)$ be defined and continuous on a closed bounded set $E$. Prove that: