Continuity of the inverse of a continuous function

[GodfreyHW]

Hey, please tell me if the following is correct.
We have a continuous, increasing and strictly monotonic function on [MATH][a, b][/MATH], and [MATH]x_0\in[a, b][/MATH]. Let [MATH]f(y)[/MATH] be its inverse, and [MATH]f(x_0)=y_0[/MATH].
I want to show that [MATH]|y-y_0|<\delta\implies|g(y)-g(y_0)|<\epsilon[/MATH].

[MATH] |g(y)-g(y_0)| < \epsilon \Leftrightarrow x_0-\epsilon<g(y)<x_0+\epsilon\\ \Leftrightarrow f(x_0-\epsilon)<y<f(x_0+\epsilon)\\ \Leftrightarrow -(y_0-f(x_0-\epsilon))<y-y_0<f(x_0+\epsilon)-y_0 [/MATH]
If I let [MATH]\delta=\min(y_0-f(x_0-\epsilon),f(x_0+\epsilon)-y_0)[/MATH], while considering small [MATH]y[/MATH]s, then I think that I have it right.
For decreasing [MATH]f(x)[/MATH], I can flip the inequality symbols at the second step, and choose [MATH]\delta=\min(f(x_0-\epsilon)-y_0,y_0-f(x_0+\epsilon))[/MATH].
Thanks!

 

[JeffM]

Wow, am I confused: g(x) is nowhere defined.

Hey, please tell me if the following is correct.
We have a continuous, increasing and strictly monotonic function on [MATH][a, b][/MATH], and [MATH]x_0\in[a, b][/MATH]

Is this function denoted y = g(x)?

Let [MATH]f(y)[/MATH] be its inverse

I presume "its" refers to g(x).

[MATH]f(x_0)=y_0[/MATH].

I thought the argument of the f function was y.

I have no idea whether I can help you if you eliminate my confusion, but I am hopeless at this stage.

EDIT: The first thing to do is prove existence.

 

[GodfreyHW]

Wow, am I confused: g(x) is nowhere defined.


Is this function denoted y = g(x)?

I presume "its" refers to g(x).

I thought the argument of the f function was y.

I have no idea whether I can help you if you eliminate my confusion, but I am hopeless at this stage.

EDIT: The first thing to do is prove existence.

Ugh, typo! I meant to write [MATH]g(y)[/MATH] for the inverse.

 

[GodfreyHW]

EDIT: The first thing to do is prove existence.

I am assuming that there is an inverse since [MATH]f(x)[/MATH] is strictly monotonic on the aforementioned interval.

 

[JeffM]

To PROVE continuity on an interval, I think you have to define the interval and prove existence at every point on that interval. In other words, before we start to mess with limits we have preliminary work to do. As I said in a different thread, I really do not love analysis. Maybe someone will come along who loves it. My son and I are talking the dog and my grandson for a walk in the woods.