Inverse function and continuity

In summary, the question is whether a continuous function that is monotonically increasing in an interval will have an inverse that is also monotonically increasing in that interval. The answer is yes, and it is actually strictly monotonically increasing for the inverse. This holds true even if the function is not continuous, as long as it is monotonically increasing in the interval.
  • #1
phymatter
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0
if a continious function is monotoniously increasing in an interval , is it necessary that its inverse will also increase monotoniously in that interval?
 
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  • #2
Wow this is a cool question. I think so.
You have: a<b then f(a)<=f(b).
You also have:f-1(f(x))=x
You're wondering: if A<B is f-1(A)<=f-1(B)

I think it might actually be strictly monotone increasing for the inverse. I'll have to think about it some more tomorrow.
 
  • #3
You don't even need "continuous". Suppose f is monotonically increasing on [a, b] but that [itex]f^{-1}(x)[/itex] is not. Then there exist u, v, in [f(a), f(b)] such that u> v but [itex]f^{-1}(u)< f^{-1}(v)[/itex]. Let [itex]p= f^{-1}(u)[/itex] and [itex]q= f^{-1}(v)[/itex]. Then we have [itex]p< q[/itex] but [itex]f(p)= u> v= f(q)[/itex] contradicting the fact that f is increasing.
 

1. What is an inverse function?

An inverse function is a function that "undoes" another function. In other words, if a function f(x) maps an input x to an output y, the inverse function f-1(y) maps the output y back to the input x. This means that the composition of a function and its inverse results in the original input.

2. How do I find the inverse of a function?

To find the inverse of a function, you can follow these steps:

  1. Write the function in the form y = f(x).
  2. Swap the x and y variables.
  3. Solve for y to get the inverse function in the form x = f-1(y).

3. What does it mean for a function to be continuous?

A function is continuous if its graph is a single unbroken curve without any holes or jumps. This means that as the input values change, the output values change smoothly and predictably. In other words, small changes in the input result in small changes in the output.

4. How can I determine if a function is continuous?

To determine if a function is continuous, you can use the following criteria:

  • The function is defined at the point in question.
  • The limit of the function as x approaches the point exists.
  • The limit and the value of the function at the point are equal.

5. Why is continuity important in mathematics?

Continuity is important in mathematics because it allows us to make precise calculations and predictions based on a function's behavior. It also allows us to generalize results and make connections between different areas of mathematics. In addition, many real-world phenomena can be modeled using continuous functions, making continuity a useful concept in various scientific fields.

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