Are the Limits of an Increasing Function the Same?

[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

 

[mathwonk]

try some simple examples, like f(x)= 0 for negative x and f(x) = 1 for non negative x.

 

[tuananh]

Define M = sup{f(x) | a <= x < c}, we prove

lim f(x) = M
x->c-

For every e>0, there exist an element a<= d < c such that

M-e < f(d) <= M

Thus, for all d < x < c, we have

M-e < f(d) <= f(x) <= M < M+e

This means

lim f(x) = M
x->c-

The second equality can be proved similarly. The two limits (left and right) are the same if the function f is continous