# Find the sup-norm

Find the sup-norm, $\|f\|_{\sup}$, if

$$f(x)=\begin{cases} 0, &x \in\mathbb{Q}\\ -x^2, &x\not\in\mathbb{Q} \end{cases}$$

As I look at the graph of $-x^2$, I know it's a decreasing function. I know the sup-norm is infinity. I'm not sure why that is though.

• This is very difficult to read. Please edit and type using Latex commands. – ADA Apr 24 '17 at 1:59

Since $$f(x) = \begin{cases} 0 & x \in \mathbb{Q} \\ -x^2 & x \notin \mathbb{Q} \end{cases},$$
we have $$|f(x)| = \begin{cases} 0 & x \in \mathbb{Q} \\ x^2 & x \notin \mathbb{Q} \end{cases}.$$
Notice that $x^2$ can get arbitrary large.
hence $$\left\| f\right\|_\infty= \sup\{|f(x)|: x\in \mathbb{R} \}= \infty$$