Question

Continuity of a Function 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.)

Short answer

The function f(x) is continuous only at x=0 and discontinuous at all other points.

Step-by-step solution

Understand continuity

A function f(x) is said to be continuous at a point a if \lim_{x \to a^+}f(x)=\lim_{x \to a^-}f(x)=f(a). This means the limit of the function as x approaches a from the left (denoted by a-) equals to the limit as x approaches a from the right (a+) and also equals to the value of the function at a.

Check the continuity at x=0

For the provided function, check at x=0. For x < 0, both rational and irrational values will get mapped to 0 since the function value is 0 for rational numbers and k*0=0 for irrational numbers. Mathematically, \lim_{x \to 0^-}f(x) = 0. Similarly, for x > 0, \lim_{x \to 0^+}f(x) = 0. And, value of function at x=0 i.e. f(0) = 0. So, \lim_{x \to 0^+}f(x)=\lim_{x \to 0^-}f(x)=f(0) = 0 and function is continuous at x=0.

Show discontinuity at x ≠ 0

At x ≠ 0, \lim_{x \to a^+}f(x) ≠ \lim_{x \to a^-}f(x) ≠ f(a) which indicates that the function is not continuous at x ≠ 0. For instance, let us consider x=1. For x > 1, if x is rational then f(x)=0 but if x is irrational then f(x)=k*x. Hence, the limits differ based on whether the x value is rational or irrational. Hence the function is not continuous at x ≠ 0.

Key concepts

Limit of a Function

In mathematics, the concept of the limit of a function is fundamental to understanding what it means for a function to be continuous. When we say the limit of a function as x approaches a certain value, we are describing how the function behaves as the input value, x, gets arbitrarily close to a point. This does not necessarily mean x will reach that point, just that it gets very close.

For example, if we are looking at the limit of f(x) as x approaches 2, we are interested in what value f(x) is approaching as x gets nearer and nearer to 2 from both sides. This is denoted as \(lim_{x \to 2}f(x)\). A limit exists if, as x approaches the value from both the left (denoted x^-) and the right (denoted x^+), the function approaches the same value.

In our exercise, understanding limits is key to analyzing the points at which the function is continuous, especially at the transition between rational and irrational numbers. Our function f(x) behaves particularly differently at rational versus irrational inputs, and it's the concept of limits that allows us to assess its continuity.

Rational and Irrational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction \(\frac{p}{q}\) of two integers, where p is the numerator, q is the denominator, and q is not zero. Examples include numbers like 1/2, 3, and -5/4.

On the other hand, irrational numbers cannot be expressed as simple fractions. They have non-repeating, non-terminating decimal expansions. Famous examples include \(pi\) and the square root of 2. Understanding the distinction between these two sets of numbers is crucial when analyzing functions that treat rational and irrational numbers differently, as in our exercise function f(x).

This distinction is also at the heart of why the function will behave discontinuously at any point other than zero. Given any interval around a non-zero point, there will always be rational and irrational numbers, causing different functional values, making it impossible to define a limit as required for continuity.

Continuous at a Point

A function is continuous at a point when, roughly speaking, you can draw its graph at that point without lifting your pen off the paper. In more precise terms, a function f(x) is continuous at a point a if the following three conditions are met:

• The function is defined at a, which means f(a) exists.
• The limit of the function as x approaches a exists.
• The limit of the function as x approaches a equals the function's value at a, f(a).

This can be summed up with the equation: \(lim_{x \to a}f(x) = f(a)\).

In the context of our problem where f(x) is defined differently on rational and irrational numbers, continuity becomes a provocative concept. The function f(x) cannot meet the continuity conditions at non-zero points because the limits from the left and right will not match up; they depend on the nature of x—whether rational or irrational. However, at zero, the function value (zero for both rational and irrational inputs) aligns with the limits from either side, satisfying the condition for continuity at that single point. This is a beautiful illustration of how a function can be continuous at some points but not everywhere.