- #1

- 32

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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

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- Thread starter dopey9
- Start date

- #1

- 32

- 0

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

- #2

mathwonk

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

- #3

- 10

- 0

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

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