Continuity of the Function f(x) x - [x]

The function #955;x x - ?x? is a critical concept in understanding the behavior of functions and their continuity. In this article, we will explore the continuity and discontinuity of this function.

Introduction

The function #955;x is a variation of the floor function, denoted as ?x?, which represents the greatest integer less than or equal to x. The function #955;x x - ?x? is known as the fractional part of x. It is a piecewise function that has different behaviors depending on the value of x.

Discontinuities of the Function

For any integer value n, the function #955;x is discontinuous. Specifically, for n ≤ x ≤ n 1, we have #955;x x - n, which simplifies to:

#955;x x - n.

For any #916; 0 1 small enough, when x is sufficiently close to n but less than n, #955;x - #916; 1 - #916;. We can write this as #955;x- 1-, as opposed to #955;x 0. Clearly, this indicates a discontinuity with a step of 1.

Interpretation of [x]

It's important to note that the notation [x] can vary. If [x] is interpreted as the absolute value of x, the answer has already been provided. However, if [x] is understood as the floor function, denoted as ?x?, the function is continuous over each interval [n, n 1) for all integers n. In this case, the limit of #955;x as x approaches n from the left is 1, whereas the limit as x approaches n from the right is 0.

Continuous Everywhere Except at Integers

Given the piecewise definition of #955;x, we can express it as:

#955;x begin{cases} 2x text{if} x leq 0 0 text{if} x > 0 end{cases}

This function is defined and continuous everywhere except possibly at x 0. To prove its continuity at x 0, we need to show that the limits from both sides equal the function value at that point:

Consider the limit as x approaches 0 from the left: limx→0- ?x 0. Evaluate #955;0 which is also 0. Consider the limit as x approaches 0 from the right: limx→0 ?x 0.

Since all these values are equal, getTypex is continuous at x 0.

Discussion

In conclusion, the function #955;x x - ?x? is continuous at every point except at integer values. This is due to the step discontinuity that occurs at each integer. Understanding the continuity and discontinuity of such functions is crucial in various mathematical contexts, including calculus and real analysis.