Do Right and Left Hand Limits Exist for All Increasing Functions?

[SomeRandomGuy]

Been bangin my head against the wall all weekend thinkin about this question:

Let f:R->R be an increasing function. Proce that lim x->c+f(x) and lim x-c-f(x) (right and left hand limits) must each exist at every point c in R.

There's more to the question, but if I can get this part solved, I'm sure the rest won't be trouble.

My original idea was to prove this by contradiction, assuming the limits don't exist, and showing this violates the increasing aspect. However, I've come to deadends each time. Proving it directly seems very difficult as well.

 

[benorin]

Has your text defined left/right-handed limits of f at x in terms of what happens to $f\left( x_n\right)$ for alll sequences that converge to x which are strictly less than/greater than x?

 

[SomeRandomGuy]

Has your text defined left/right-handed limits of f at x in terms of what happens to $f\left( x_n\right)$ for alll sequences that converge to x which are strictly less than/greater than x?

Yes,

limx->c+ f(x) = L if limf(x_n)=L for all x_n > c where lim(x_n)=c.

Similar definition for for limx->c- f(x).

Since f was defined as an increasing function, then x_n is either increasing or decreasing depending on what limit we are taking. How can we conclude that limf(x_n) = L? Still kind of confused. I'll go work on it and check back in the morning.

Thanks for the help.

 

[benorin]

This PDF pg. 182 [pg. 194 of the PDF] Theorem 1.