function continuous at only one point

Let us show that the function f:,

f(x)={x,when x is rational,-x,when x is irrational,

is continuous at x=0, but discontinuousMathworldPlanetmath for all x{0} [1].

We shall use the following characterization of continuity for f: f is continuous at a if and only if limkf(xk)=f(a) for all sequences (xk) such that limkxk=a.

It is not difficult to see that f is continuous at x=0. Indeed, if xk is a sequence converging to 0. Then

limk|f(xk)| = limk|f(xk)|
= limk|xk|
= 0.

Suppose a0. Then there exists a sequence of irrational numbers x1,x2, converging to a. For instance, if a is irrational, we can take xk=a+1/k, and if a is rational, we can take xk=a+2/k. For this sequence we have

limkf(xk) = -limkxk
= -a.

On the other hand, we can also construct a sequence of rational numbersPlanetmathPlanetmath y1,y2, converging to a. For example, if a is irrational, this follows as the rational numbers are dense in , and if a is rational, we can set yk=xk+1/k. For this sequence we have

limkf(yk) = limkyk
= a.

In conclusionMathworldPlanetmath f is not continuous at a.


  • 1 Homepage of Thomas Vogel, tom.vogel/gallery/node3.htmlA function which is continuous at only one point.
Title function continuous at only one point
Canonical name FunctionContinuousAtOnlyOnePoint
Date of creation 2013-03-22 14:56:19
Last modified on 2013-03-22 14:56:19
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 7
Author Andrea Ambrosio (7332)
Entry type Example
Classification msc 26A15
Classification msc 54C05
Related topic DirichletsFunction
Related topic FunctionDifferentiableAtOnlyOnePoint