Can We Prove a Function with Intermediate Value Property is Continuous at x?

[vidyarth]

Let \(\displaystyle f\) be a function with the intermediate value property. In addition, let it have the property that \(\displaystyle |f(x)-x_n|\le M\cdot sup_{n,m}|f(x_n)-f(x_m)|\), where \(\displaystyle M\) is a constant and \(\displaystyle x_n\) is a sequence converging to \(\displaystyle x\). Then, can we show that \(\displaystyle f\) is continuous? I think we have to tackle this problem by considering a neighbourhood of \(\displaystyle x\) and then use Cauchy criterion for convergence. Any ideas? Thanks beforehand.

 

[jvicens]

Hello,

Thank you for bringing up this interesting problem. I believe that your approach of using the Cauchy criterion for convergence is a good starting point. However, I think we can also use the intermediate value property to show that f is continuous at x.

First, let's consider a neighborhood of x, denoted as N(x). By the intermediate value property, we know that for any y\in N(x), there exists a z\in N(x) such that f(z)=y. This means that f is continuous at x, since we can always find a point in the neighborhood that maps to any desired output.

Now, let's focus on the given inequality: |f(x)-x_n|\le M\cdot sup_{n,m}|f(x_n)-f(x_m)|. We can rewrite this as |f(x)-f(x_n)|\le M\cdot sup_{n,m}|f(x_n)-f(x_m)|+|x-x_n|. This shows that as x_n approaches x, the difference between f(x) and f(x_n) also approaches 0. This is a key property of continuity, as it means that the function values are getting closer and closer to each other as the input values get closer.

Combining these two observations, we can conclude that f is continuous at x. We have shown that for any neighborhood of x, we can find a point in that neighborhood that maps to any desired output, and that the function values get closer as the input values get closer. This satisfies the definition of continuity, and thus we can say that f is continuous at x.

I hope this helps in your understanding of the problem. Let me know if you have any further thoughts or ideas. Thank you.