# What does my textbook mean by Hypothesis here?

I am reading through page 278 of the textbook "Calculus: Early Transcendentals" 8th Edition, by James Stewart. I am confused by his use of the word "hypothesis" here. I thought a hypothesis was essentially an educated guess? What does the author mean by "hypothesis" here?

• The assumptions of the theorem. Mar 21, 2019 at 20:18
• Hypothesis here means the assumptions of the theorem, the 'if' part. For example, if the theorem goes, "If $f$ is a continuous function on a closed interval, then...", the hypothesis is the assumption of continuity and closed interval. Mar 21, 2019 at 20:22
• The theorem assumed certain things (continuity and closed interval). The author is referring to these, saying how the theorem does not hold anymore if one of these assumptions is omitted. Mar 21, 2019 at 20:22
• The author wants to say that functions not satisfying the assumptions of the theorem may not have absolute maximum/minimum, whilst the ones that do will necessarily have these extrema. Mar 21, 2019 at 20:27
• Any theorem is of the form "If A then B". The "hypothesis" is "A". That has nothing to do with a guess, "educated" or not! Mar 21, 2019 at 20:28

Hypothesis: Let $$f$$ be a continuous, real-valued function to the closed interval $$[a,b]$$
Conclusion: Then $$f$$ attains extreme values on that interval
• Yes exactly. More rigorously, extreme values would be defined like this for Extreme Value Theorem: there exists $c,d$ in $[a,b]$ such that $f(c)\leq f(x) \leq f(d)$ for all $x$ in $[a,b]$ Mar 21, 2019 at 20:37