Problem 32 - College Algebra and Calculus: An Applied Approach

Question

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=x-\llbracket x \rrbracket\)

Step-by-Step Solution

Answer

The function \(f(x)=x - \lfloor x \rfloor \) is continuous on the interval of all real numbers except integers. There are discontinuities at all integers. This is due to the definition of the greatest integer function, which changes value at each integer, causing a break in the graph of the function.

Step 1: Understanding the function

This is a piecewise function which can also be written as \(f(x) = 0 \) for \(x \in Z \) (all integers), and \( f(x) = x \) for \(x \in R-Z \) (all real numbers except integers). This function depicts the fractional part of x.

Step 2: Identify the discontinuities

It's clear the function has discontinuities at all integers. It is because, the left limit at any integer 'n' is 1 and the right limit at 'n' is 0, which obviously don't match and hence the function is not continuous at integer points. In mathematical terms, \(\lim_{{x \to n^-}} f(x) = 1 \) and \(\lim_{{x \to n^+}} f(x) = 0 \) for all \(n \in Z\), and obviously, \(1 \ne 0\).

Step 3: Zones of continuity

Between the integers, the function is equivalent to f(x) = x, a linear function, which is continuous for all real numbers. So, it is continuous between each pair of integers.

Key Concepts

Piecewise Functions

Piecewise functions are fascinating because they are defined by different expressions or rules over different intervals. This kind of function can switch its expression depending on the input, and that's precisely why they are called "piecewise". A piecewise function is given in segments like steps on a staircase, where each step functions under its own rule. In our original exercise, we have the function:

• For integer values of \(x\), such as \(...,-1, 0, 1, 2,...\), the function \(f(x)\) equals zero: \(f(x) = 0\).
• For non-integer real numbers, the function takes on the value of \(x\) itself: \(f(x) = x\).

Understanding how to break down piecewise functions is crucial because each segment could potentially bring different properties, like continuity or discontinuity, which we'll explore further.

Discontinuities

In mathematics, discontinuities occur when a function jumps or interrupts its behavior at a certain point. For the given function in our exercise, discontinuities are quite evident at integer points. Consider how the function behaves:

• As you approach an integer from the left (say moving towards \(n\)), the left-hand limit \(\lim_{{x \to n^-}} f(x) = 1\).
• As you approach the same integer from the right, \(\lim_{{x \to n^+}} f(x) = 0\).

These values don't match, hence causing a jump at each integer point. Such mismatched left and right limits are a classic sign of discontinuity, signaling that our piecewise function has distinct behavior at these points.

Limit and Continuity Conditions

Understanding the conditions for continuity revolves around how well the function behaves at a point. For a function to be continuous at a point \(c\), three conditions must be satisfied:

• The function \(f(x)\) must be defined at \(c\).
• The limit \(\lim_{x \to c} f(x)\) must exist.
• The limit must equal the value of the function at that point, \(\lim_{x \to c} f(x) = f(c)\).

Given our original function, these conditions fall apart at integers since the left and right limits differ. This difference means the second condition isn't fulfilled at these points, causing a lack of continuity. In contrast, between any two integers, the function behaves linearly following \(f(x) = x\), perfectly upholding all conditions for continuity and ensuring the function runs smoothly without interruptions.