increasing/decreasing function?

[apple2357]

Ok, all this is making sense to me now. Thanks.
BTW, this is a photo from the textbook of the Introduction to the topic:



This felt like definitions to me rather than a theorem.

 

[Dr.Peterson]

Yes, that looks like the sort of slightly deficient introduction I expected to find. They have leaped over the actual definition (and even, apparently, an informal description of what "increasing" looks like in a graph) to the calculus-based characterization that results from a theorem.

But I observe that they have not lied! They say that IF the derivative is positive everywhere, THEN the function is increasing. That's true! They didn't say it can't be increasing under other conditions (though it is natural to read this as if it were a definition, identifying all conditions).