does that mean that "if f has no absolute max on [0,1] then f is discontinuous" is true?
Not necessarily. As I mentioned earlier, you need only one counterexample to prove something false, but proving something true is much more difficult. No matter how many examples you can find that satisfy the criteria, that's
never enough on its own. Although, I do need to give a small correction to my previous post, because I learned something new just today. The book I learned from defined a global maximum as:
f(x) has an global maximum at the point x= c, if for every x in the domain of f(x),
f(x)<f(c)
This allows for one and only one global maximum, such that the constant function y = 5 would not have a global maximum. But I found out that this definition is apparently not the typical definition. Most books instead allow for multiple global maxima by using the criteria
f(x)≤f(c). Let this serve as a lesson at my expense that terminology and definitions matter a great deal.
I know that the extreme value theorem states that if f is continuous on a closed interval [a,b] then f attains both an absolute max and an absolute min.
if there is no absolute max, does that mean the function must therefore be discontinuous?
Again, this really depends on exactly how we define the terms involved. Take, for example, the function x
2. It is continuous on the closed interval [2, 5], so this formulation of the extreme value theorem says it ought to have a global maximum somewhere in this interval. But merely the fact that the function is continuous on this interval doesn't guarantee the existence of said maximum. The function "blows up" to infinity at either extreme, so it clearly has no global maxima. However, if we
restrict the domain to be [2, 5], then it does have a global maximum.
So really, it all comes down to how do you read and interpret the problem. Whether or not the statement is true depends on whether we're looking at the function as a whole or specifically restricting the domain to [0, 1]. And only the person who wrote the problem can tell you that for sure. Perhaps your instructor has some guidance on how they'd interpret it?