increasing funtions

[dopey9]

A function f : A ->R is increasing if f(x) <= f(y) for every x, y in A such that x <= y.

Suppose that f : [a, b] -> R is increasing and that a < c < b.

i want to shat that :
lim f(x) = sup{f(x) | a <= x < c} and
x->c-

limf(x) = inf{f(x) | c < x <= b}.
x->c+

and whether these limits are the same?

can anyone help with this

 

[pka]

Consider the set \(\displaystyle L = \left\{ {f(x):x \in [a,c)} \right\}\). Then L is bounded above by \(\displaystyle f(c)\).
Thus L has a least upper bound, say \(\displaystyle {\rm B} = \sup (L)\).
So, \(\displaystyle \varepsilon > 0\quad \Rightarrow \quad \left( {\exists f(t) \in L} \right)\left[ {{\rm B} - \varepsilon < f(t) \le {\rm B}} \right]\).
But that is the definition of \(\displaystyle \lim _{x \to c^ - } f(x)\).

Now you should be able to complete the proof.