Continuous at Only One Point Show that the following...

Continuous at Only One Point Show that the following function is continuous only at $x=0$ :
$$f(x)=\left\{\begin{array}{ll}{x} & {\text { for } x \text { rational }} \\ {-x} & {\text { for } x \text { irrational }}\end{array}\right.$$

Key Concepts

Density of Rational and Irrational Numbers

The rational and irrational numbers are both dense in the real numbers, meaning that in any open interval around any real number, there are both rational and irrational numbers. This property is crucial when analyzing functions defined separately on the rationals and irrationals, as it forces the function to take values from both definitions arbitrarily close to any point.

Limit of a Function

The concept of a limit is central to understanding continuity. It involves investigating the behavior of the function as the input values approach a particular point. When dealing with functions that have separate definitions on different subsets of the domain, one must consider the limits along sequences from each subset to determine if a unique limit (and hence continuity) exists.

Continuity of a Function

Continuity at a point means that the limit of the function, as the input approaches that point from any direction, exists and equals the function's value at that point. This is typically checked using the epsilon-delta definition of continuity, where for every positive epsilon there is a corresponding delta that ensures the function's output is within epsilon of the value when the input is within delta of the point.

Piecewise-Defined Functions

Piecewise-defined functions are those that are described by different expressions on different parts of the domain. Analysis of such functions often requires examining the behavior at the boundaries or switching points between the different expressions to ascertain properties like continuity or differentiability.

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