Are the Limits of an Increasing Function the Same?

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In summary, a function is considered increasing if its output is always greater than or equal to the output of any input that comes before it. Given an increasing function f : [a, b] -> R where a < c < b, we can prove that lim f(x) = sup{f(x) | a <= x < c} and x->c- lim f(x) = inf{f(x) | c < x <= b}. These two limits will be equal if the function is continuous at point c.
  • #1
dopey9
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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
 
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  • #2
try some simple examples, like f(x)= 0 for negative x and f(x) = 1 for non negative x.
 
  • #3
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
 

Related to Are the Limits of an Increasing Function the Same?

1. What is an increasing function?

An increasing function is a mathematical concept where the output value of a function increases as the input value increases. This means that as the independent variable increases, the dependent variable also increases.

2. How can I identify an increasing function?

To identify an increasing function, you can look for a positive slope on a graph of the function or check if the first derivative of the function is positive for all values of the independent variable. You can also check if the function is monotonic increasing, meaning it does not decrease at any point.

3. What is the difference between a strictly increasing and a non-decreasing function?

A strictly increasing function is a type of increasing function where the output value must increase by a distinct amount for every increase in the input value. On the other hand, a non-decreasing function is an increasing function where the output value can remain constant for some intervals of the input value.

4. Can an increasing function have negative values?

Yes, an increasing function can have negative values. The key characteristic of an increasing function is that the output value increases as the input value increases, regardless of the sign of the values.

5. How can increasing functions be useful in science?

Increasing functions are commonly used in various scientific fields such as physics, economics, and biology. They can be used to model relationships between variables and make predictions about how one variable will change as another variable increases. They are also useful in analyzing trends and patterns in data.

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