Reading Haaser-Sullivan's Real Analysis

[_DJ_british_?]

Hi peeps!

I was reading Haaser-Sullivan's Real Analysis and came across a problem for which I have a doubt. A part of it is stated like this : " For all x in the closed interval [a,b] in R, |g'(x)|<=1 '' (g(x) is, of course, a real-valued function of a real variable and that's all we know about it). Does that mean that for all x in [a,b], g'(x) is defined or that for all x in [a,b] such that g'(x) is defined, |g(x)|<=1?

Thanks in advance!

 

[HallsofIvy]

I would interpret it as saying that g'(x) is defined and between -1 and 1, for all x in [a,b].

 

[_DJ_british_?]

That's what I thought, thanks! I haven't touched the formal definition of differentiability, but in Calculus I learned that a function was differentiable on [a,b] iff its derivative exists on [a,b]. So the condition stated above is enough to show differentiability on [a,b] and thus, continuity and contraction?

 

[Mark44]

That's what I thought, thanks! I haven't touched the formal definition of differentiability, but in Calculus I learned that a function was differentiable on [a,b] iff its derivative exists on [a,b]. So the condition stated above is enough to show differentiability on [a,b] and thus, continuity and contraction?

I think that all we can say is that since |g'(x)| is defined at every x in the interval, then |g(x)| is continuous on the same interval, but that g(x) is not necessarily continuous.

What do you mean by "contraction?" Are you saying that |g(x)| <= x?