Thread: Supremum and maximum

Can anyone provide an example of a differentiable function f(x) such that a supremum of f exists but a maximum of f does not exist, for $\displaystyle 0\leq x \leq k$ (k is any fixed number)?

Thanks very much.

Originally Posted by WWTL@WHL

Can anyone provide an example of a differentiable function f(x) such that a supremum of f exists but a maximum of f does not exist, for $\displaystyle 0\leq x \leq k$ (k is any fixed number)?

Thanks very much.

Wait, this question doesn't make sense. It is wrong. Are you sure that it's the right question?

It's not a question I've been set, but it's just something I've been thinking about.

So I guess you're telling me there is no such function? Could you kindly explain why?

Originally Posted by WWTL@WHL

It's not a question I've been set, but it's just something I've been thinking about.

So I guess you're telling me there is no such function? Could you kindly explain why?

The real reasoning? Continuous(and thus differentiable) functions preserve compactness. Or, more simply Extreme value theorem - Wikipedia, the free encyclopedia