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Chapter 5: Problem 7

Give an example of a function \(f:[0,1] \rightarrow \mathbb{R}\) that is discontinuous at every point of \([0,1]\) but such that \(|f|\) is continuous on \([0,1]\).

Short Answer

Expert verified
The function \(f(x) = (-1)^{\lfloor 1/x \rfloor}\) when \(x \neq 0\) and \(f(x) = 1\) when \(x = 0\) is discontinuous at every point in the interval [0,1] but its absolute value is continuous on the same interval.

Step by step solution

01

Concept behind the Function Selection

The key to this problem is realizing that to make a function discontinuous at every point in the interval, we have to ensure the function flips its sign at every point. That means, to move from one point to the other in the interval, the function must jump from a negative to a positive value, or vice versa.
02

Constructing the Function

The function \(f(x) = (-1)^{\lfloor 1/x \rfloor}\) is discontinuous at every rational point, because when x approaches a rational number r (in lowest terms), 1/x will approach either \( \pm \infty \) and the function will 'jump' from -1 to 1 or from 1 to -1 and therefore discontinuous at r. However, it is undefined at x=0. But since the problem domain is from [0,1], adjust it slightly to \(f(x) = (-1)^{\lfloor 1/x \rfloor}\) for \(x \neq 0\) and \(f(x) = 1\) for \(x = 0\).
03

Testing the Function

We can check if this function satisfies the requirements. It is discontinuous at every point of (0,1] since it flips its sign at every rational number r. Additionally, the function satisfies \(f(0) = 1\), ensuring continuity at 0.
04

Checking the Absolute Function

Now we need to verify that the absolute value of the function is continuous. The absolute value of the function \(|f(x)|\) will flip all the negatives to positive, aligning them with the other half. This means the function \(|f(x)|\) is indeed continuous over [0,1].

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