What's the difference between Boundedness Theorem and Extreme Value Theorem?

Question

I know that the Boundedness Theorem states that if a function is continuous on a closed interval then it is bounded on that interval, but doesn't it mean that it also attains the maximum and minimum values? How can a function be bounded on a interval, but at the same time not attain the maximum and minimum?

Answer 1

Take, for instance, $f\colon[-1,1]\longrightarrow\mathbb{R}$ defined by $$ f(x)=\begin{cases}x&\text{ if }x\in(-1,1)\\0&\text{ if }x=\pm1,\end{cases} $$ Then $f$ is bounded, but it doesn't attain a maximum or a minimum.