3. three key examples: (a) give an example of a continuous function on a closed, unbounded domain which does…

Explanation:

Step1: Recall extreme - value theorem

The extreme - value theorem states that a continuous function on a closed and bounded interval $[a,b]$ attains its absolute maximum and minimum.

Step2: Answer part (a)

Consider the function $f(x)=x$ on the unbounded closed domain $[0,\infty)$. Since the domain is unbounded, as $x$ goes to $\infty$, there is no absolute maximum. So an example is $y = x$ on $[0,\infty)$.

Step3: Answer part (b)

Consider the function $f(x)=\frac{1}{x}$ on the bounded domain $(0,1]$. The function is continuous on $(0,1]$, but as $x$ approaches $0$ from the right, $f(x)$ goes to $\infty$. So it does not achieve an absolute maximum on the non - closed (open on the left) bounded domain $(0,1]$.

Step4: Answer part (c)

Consider the function $f(x)=\begin{cases}x, &x\in[0,1)\0, &x = 1\end{cases}$ on the closed and bounded domain $[0,1]$. This function is discontinuous at $x = 1$. As $x$ approaches $1$ from the left, $f(x)$ approaches $1$, but $f(1)=0$. So it does not achieve an absolute maximum.

Answer:

(a) $y = x$ on $[0,\infty)$ (b) $y=\frac{1}{x}$ on $(0,1]$ (c) $f(x)=\begin{cases}x, &x\in[0,1)\0, &x = 1\end{cases}$ on $[0,1]$