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?

 

[CellCoree]

Hello,

I am glad to hear that you are reading Haaser-Sullivan's Real Analysis and have come across a problem that has sparked your curiosity. Based on the information given, it seems that the statement "for all x in the closed interval [a,b] in R, |g'(x)|<=1" is referring to the behavior of the derivative of g(x) within the interval [a,b]. This means that for all values of x within that interval, the absolute value of the derivative of g(x) is less than or equal to 1. This does not necessarily mean that g'(x) is defined for all values of x in [a,b], but rather that when it is defined, its absolute value is bounded by 1. I hope this helps clarify your doubt. Keep up the good work in your studies!