1. increasing functions

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...any guidance..or starting me off thankz

2. Originally Posted by dopi
and whether these limits are the same?
I do not think so, for the functions can be discontinous.

3. Is this what i need to use

The set L={f(x): x in[a,b]]

Then L is bounded above by . f(c)

Thus L has a least upper bound, say B= sup{L}
So, . epsilon >0 => (there exists f(t) in L)[B-epsilon<f(t)<= B]
this is the definition of lim f(x( as x->c ...this is what i got from my class notes...is this what i could use to prove solve the question i posted before...can anyone show me please thankz

4. Originally Posted by dopi
The set L={f(x): x in[a,b]]

Then L is bounded above by . f(c)

Thus L has a least upper bound, say B= sup{L}
So, . epsilon >0 => (there exists f(t) in L)[B-epsilon<f(t)<= B]
this is the definition of lim f(x( as x->c ...this is what i got from my class notes...is this what i could use to prove solve the question i posted before...can anyone show me please thankz
I think you should say,
$\displaystyle \{ f(x)|x \in [a,c)\}$ is bounded by $\displaystyle f(c)$ because the function is increasing. Then by Bolzano-Weirestass theorem $\displaystyle \lim_{x\to c^-}f(x)$ exist, also it is the least upper bound that is the supremum of the sequence. Similar argument for the other half.