Real Analysis - Continuity

Question

a. Give an example of a function defined everywhere on the interval $[0,1]$, which does not achieve its maximum.

b. Give an example of a function defined on $\mathbb{R}$, that is nowhere continuous.

c. Give an example of a continuous function defined on a bounded set, which is not uniformly continuous.

My answers are

1. $f(x) = x^2$
2. $f(x) = \{1$, if $x$ is rational; $-1$, if $x$ is irrational$\}$
3. $f(x) = 1/x$

My workings are attached. Please help verify if the working is correct.Solutions

Answer 1

$a)$ Consider $f(x)$ on $[0,1]$ such that $f(0) = 0, f(1) = 0, f(x) = \dfrac{1}{x}, 0 <x < 1$.

$b)$ The function you had is a good one.

$c)$ Consider $f(x)$ on $(0,1)$ and $f(x) = \dfrac{1}{x}$.

Answer 2

1. Wrong: the maximum is $1$, which is $f(1)$.
2. Right.
3. Your answer is incomplete, at best, since you did not state what is the domain of your function.

Answer 3

1)

Extreme value theorem says that every continuous function over a closed interval has a maximum.

You need a function that is not continuous.

$f(x) = \begin{cases} f(x) = x & x<1\\f(x) = 0 & x=1 \end{cases}$