Proving Continuity of Monotone Functions on Interval Domains

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In summary, the conversation discusses how to prove the continuity of a monotone function f on a set A, given that the range of f is an interval. The discussion includes using a picture to understand the conclusion, finding a consistent analytical proof, and using the fact that the range is an interval to show that a jump discontinuity cannot occur. The conversation also mentions using the result to prove the continuity of inverse functions, but notes that the inverse of a continuous, monotone function with an interval range may still have a jump discontinuity. An example is given to illustrate this point.
  • #1
math8
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Suppose f:A-->R is monotone (ACR: reals)
and suppose the range of f is an interval, show f is continuous on A.

By drawing a picture, I can see the conclusion. Since f is monotone, the only type of discontinuity it may have is a jump discontinuity. But since the range of f is an interval, this cannot happen.

I would like to have a more consistent proof (analytical proof), I mean, I know that since f is monotone, for each point a of A, f(a+) and f(a-) exist. Now f(a+)> or eq. to f(a) and f(a-)< or eq. to f(a). Now how would I show that f(a+)< or eq. to f(a) and f(a-)> or eq. to f(a) using the fact that range(f) is an interval? Hence I would be able to conclude that f(a-)= f(a+)=f(a), therefore, f continuous on A.
 
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  • #2
Just take the direct approach. Pick an x and an epsilon e>0. You want to show that there is a delta d>0 such that |f(x)-f(y)|<e if |x-y|<d, right? Think about x=f^(-1)(f(x)), f^(-1)(f(x)+e) and f^(-1)(f(x)-e). Can you figure out how to define a d using those?
 
  • #3
I would use d such that |f^(-1)(f(x)+e) - f^(-1)(f(x)-e)|> d. Is that correct? How do we use the fact that range of f is an interval?
I see that you're using f^(-1), in fact the problem has a subproblem that says to use the result to prove the continuity of inverse functions. How do we conclude that?
 
  • #4
If there is a jump discontinuity, then the range isn't an interval. I use that the range is an interval to show that a point like f^(-1)(f(x)+e) even exists. You've probably drawn a picture of a function with a jump discontinuity, so you should be able to see how the proof breaks down. I would say d should be smaller than the maximum of |f^(-1)(f(x)+e)-x| and and |f^(-1)(f(x)-e)-x|. Do you see why? Try doing the subproblem about f^(-1) and see how far you get.
 
  • #5
shouldn't d be smaller than the minimum of |f^(-1)(f(x)+e)-x| and |f^(-1)(f(x)-e)-x| instead?
 
  • #6
Sure. Absolutely. I was just testing you. :)
 
  • #7
oh, good, then, at least I understand what's going on...
 
  • #8
That's what I was checking. Actually 'maximum' was just a typo...
 
  • #9
I am trying for the f^-1 part, but I cannot get anywhere, can you give me another hint?
 
  • #10
What exactly is the f^(-1) part?
 
  • #11
the subproblem that says to use the result to prove the continuity of inverse functions.
 
  • #12
The inverse of a continuous function is not necessarily continuous even if it's monotone and the range is an interval. Can you state the full problem? Do you know something about the domain?
 
  • #13
that's it, that's how it is stated:

"Suppose f:A-->R is monotone (ACR: reals)
and suppose the range of f is an interval, show f is continuous on A. Use this result to prove the continuity of inverse functions".
 
  • #14
Let f(x)=x for x in [0,1) and f(x)=x-1 for x in [2,3]. f is continuous. The domain is [0,1)U[2,3] which is a subset of the reals. The range is [0,2]. It's monotone. The inverse has a jump discontinuity at 1. Do you see my problem?
 
  • #15
yeah, I guess, there is a problem or maybe something missing in the question, anyways, thanks for your help.
 

FAQ: Proving Continuity of Monotone Functions on Interval Domains

What is monotonicity?

Monotonicity refers to the property of a function where its output consistently increases or decreases as the input increases. In other words, if the input values are arranged in increasing order, the corresponding output values will also be arranged in increasing order.

What is the difference between monotonicity and continuity?

Monotonicity and continuity are related but distinct concepts. Monotonicity is a property of a function, while continuity is a property of a point on a function's graph. A function can be monotonic without being continuous, and vice versa.

How is monotonicity related to the derivative?

The derivative of a function measures its rate of change at a given point. A function is monotonic if and only if its derivative is either always positive or always negative. This means that the function is either always increasing or always decreasing.

Can a function be monotonic and discontinuous?

Yes, a function can be monotonic and discontinuous. For example, the function f(x) = x^2 is monotonic but has a discontinuity at x = 0. This is because it is always increasing, but the limit of the function at x = 0 does not exist.

How do you determine if a function is continuous?

A function is continuous at a point if the limit of the function at that point exists and is equal to the function's output at that point. A function is continuous on an interval if it is continuous at every point within that interval. This means that there are no breaks, jumps, or holes in the graph of the function.

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