- #1

James LeBron

- 23

- 0

## Homework Statement

Give an example of a non-continuous function on [0,1] that has no maximum and no minimum.

## Homework Equations

Well, a continuous function on a non-empty compact set will have a maximum and a minimum, so I guess this is why we need a non-continuous function.

## The Attempt at a Solution

Does this work?

f(x) =

1/2 if x is a rational number

x if x is an irrational number.

Does this work? Breaking this up into 1/2s (if rational) and x's (if irrational) hopefully makes it non-continuous, and I suppose there is always some larger number that could be formed as we approach 1 (so no maximum). What do you think?