# Prove function continuous at only one point

• cmajor47
In summary, the conversation discusses how to prove that the function f(x)= x when x is rational and -x when x is irrational is only continuous at 0. The writer mentions a website that provides a proof for this and asks for clarification on certain aspects of the proof. The key concept is that a function is continuous at a point if, for any sequence of numbers that approach that point, the function values also approach that point. To show that f is not continuous at a, one only needs to find one counter-example where the limit of the function values does not equal the value of the function at that point. The writer concludes by asking if this statement is correct and summarizes the main idea behind the proof.
cmajor47

## Homework Statement

Prove that the function defined as f(x)= x when x is rational and -x when x is irrational is only continuous at 0.

## The Attempt at a Solution

I have been looking at this website which proves this:
http://planetmath.org/encyclopedia/FunctionContinuousAtOnlyOnePoint.html

I don't understand how the writer proves that the function is not continuous when a$$\neq$$0.
Why do they take xk= to something? How does this help them show that the limits equal a and -a.
How does showing that prove that f is not continuous at a?

I know those are a lot of question, but maybe an answer to even just the first ones would help me understand this proof.

Last edited by a moderator:
Do you agree that
"f is continuous at a, if for all sequences xk that converge to a, f(xk) converges to f(a)" ?

So then to show that f is not continuous at a, all you need is to find one sequence that does converge to a but for which the limit of the f(xk) is not f(a). (The negation of "for all sequences S, P(S)" is "there exists a sequence S such that not P(S)").

So what they do is take two different sequences which both have limit a, and show that when you apply f first and take the limit, you get two different results. Since f(a) cannot be equal to a and -a at the same time, at least one of this sequence forms a counter-example to the statement of continuity they start out with.

## What does it mean for a function to be continuous at only one point?

When a function is continuous at only one point, it means that the function is continuous at that specific point but not at any other point in its domain. This means that the function has a discontinuity at all other points besides the one specified.

## How can you prove that a function is continuous at only one point?

In order to prove that a function is continuous at only one point, you must first determine the point at which the function is continuous. Then, you must show that the limit of the function at that point exists and is equal to the value of the function at that point. Finally, you must show that the function is not continuous at any other point in its domain.

## Can a function be continuous at more than one point?

Yes, a function can be continuous at more than one point. In fact, most functions are continuous at multiple points in their domain. A function is only considered to be continuous at only one point if it is not continuous at any other point in its domain.

## What are some common examples of functions that are continuous at only one point?

Some common examples of functions that are continuous at only one point include the absolute value function, the floor function, and the ceiling function. These functions have discontinuities at all other points in their domain besides the specified point.

## Why is it important to understand when a function is continuous at only one point?

Understanding when a function is continuous at only one point can help us better understand the behavior of the function. It can also help us identify and analyze any potential discontinuities in the function, which can be important in many real-world applications and mathematical proofs.

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