Show that the function f(x)={ 0, if x is rational k x, if x is irrational . i

step 1 So here we're looking at a function that we determine if it is continuous at x equals zero. So b

Show that the function f(x)={ 0, if x is rational ...

Show that the function
$f(x)=\left\{\begin{array}{ll}{0,} & {\text { if } x \text { is rational }} \\ {k x,} & {\text { if } x \text { is irrational }}\end{array}\right.$
is continuous only at $x=0 .$ (Assume that $k$ is any nonzero real number.)

Key Concepts

Sequential Characterization of Limits

The sequential approach to limits involves analyzing the behavior of the function along sequences that approach a specific point. This characterization states that a function is continuous at a point if the limits along every sequence converging to that point produce the same result as the function’s value at the point. This perspective is particularly useful when dealing with functions that have different definitions on different subsets of the domain.

Density of Rational and Irrational Numbers

In the context of real numbers, both rational and irrational numbers are dense, meaning that in any interval, no matter how small, there are both rational and irrational numbers. This density property is crucial when analyzing functions defined differently on these sets because it allows the construction of sequences from both rationals and irrationals converging to the same point, thereby enabling a comparison of the function’s behavior on these two intertwined subsets.

Continuity

Continuity refers to the property of a function where the function's value at a point is equal to the limit of the function as the input approaches that point. In mathematical terms, a function is continuous at a point if for every sequence converging to that point, the function values converge to the function value at that point. This concept is fundamental in calculus and real analysis and is central to understanding limits and differentiability.

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