# Doubt with a theorem that if f is cont. at interval it has max

• Hernaner28
In summary, the conversation discussed a theorem stating that if a function is continuous in an interval [a,b], then it has a maximum value. The demonstration of this theorem involves assuming an absurd scenario where the function never reaches its supremum value. The purpose of this assumption is to prove the theorem by contradiction.
Hernaner28
I THINK this theorem in short words states that if a function is continuous in an interval [a,b] then it has a maximum. But I have a doubt with the demonstration when it supposes an absurd:

\begin{align} & \text{THEOREM:} \\ & \text{ }f:[a,b]\to \mathbb{R}\text{ continuous} \\ & \text{then }\exists c\in [a,b]:f(c)\ge f(x)\forall x\in [a,b]\text{ } \\ & \text{ } \\ & \text{Dem}\text{. } \\ & \text{We know that }\exists \alpha =\sup f([a,b])\text{. } \\ & \text{Suppose as and absurd that }\forall x\in [a,b]\text{ }f(x)<\alpha \Rightarrow \alpha -f(x)>0 \\ & \text{ }...\text{continues}... \\ \end{align}

What I don't understand is... if alpha is sup then is the least upper bound, so wouldn't the absurd be that f(x) is GREATER than alpha, i.e. greater than a supposed number which we know is supremum?Thanks!

If you start by assuming that f(x) is larger than alpha you assumed something that was false independent of anything to do with the theorem. The point is to assume something that could be true if it wasn't for the theorem you are trying to prove. In this case you are assuming that f never takes on the value of alpha, something which can happen in general but cannot happen in this specific situation

Ah I get it now! Thank you very much!

## What is the theorem about continuity and maximum values?

The theorem states that if a function f is continuous at a closed interval [a,b], then it must have a maximum value within that interval.

## How can we prove this theorem?

This theorem can be proven using the Extreme Value Theorem, which states that a continuous function on a closed interval must have both a maximum and minimum value within that interval.

## What is the significance of this theorem?

This theorem is significant because it allows us to determine the maximum value of a function on a given interval without having to evaluate every point in that interval. It also helps us to understand the behavior of continuous functions.

## Can this theorem be applied to all continuous functions?

Yes, this theorem can be applied to all continuous functions. As long as a function is continuous on a closed interval, it will have a maximum value within that interval.

## Are there any exceptions to this theorem?

There are no exceptions to this theorem. As long as the function is continuous on the given interval, it will have a maximum value within that interval.

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