Question

Let \([x]\) denote the largest integer contained in \(x\), that is, \([x]\) is the integer such that \(x-1<[x] \leq x ;\) and let \((x)=x-[x]\) denote the fractional part of \(x .\) What discontinuities do the functions \([x]\) and \((x)\) have?

Short answer

Answer: Both functions \([x]\) and \((x)\) are discontinuous at all integer values of \(x\), that is, at all \(x \in \mathbb{Z}\).

Step-by-step solution

Analyzing the behavior of \([x]\) function

To analyze the behavior of \([x]\), we need to identify at what points the function can be discontinuous. Based on the definition of the function, we know that \([x]\) jumps to the next integer as soon as \(x\) reaches that integer. In other words, there will be a discontinuity in the function every time \(x\) is an integer. Example: When \(x\) moves from \(2.9\) to \(3\), \([x]\) jumps from \(2\) to \(3\).

Analyzing the behavior of \((x)\) function

Now let's analyze the behavior of the \((x)\) function. We see that the function is defined as \((x) = x - [x]\), which means it represents the fractional part of \(x\). We can see that as the fractional part of \(x\) increases and reaches \(1\), it drops back to \(0\). This indicates that there will be a discontinuity at every integer value of \(x\). Example: When \(x\) moves from \(2.9\) to \(3\), \((x)\) jumps from \(0.9\) to \(0\).

Identifying the discontinuities

Based on our analysis, we can conclude that both functions \([x]\) and \((x)\) have discontinuities at integer values of \(x\). That is, both functions are discontinuous at all \(x \in \mathbb{Z}\) (all integer values).

Key concepts

Greatest Integer Function

The greatest integer function, also known as the floor function, is denoted by \( [x] \) and it represents the largest integer less than or equal to a given number \( x \). This function appears as a step-like graph, where each 'step' corresponds to an integer value. To visualize, imagine walking up a staircase; as you reach the end of one step and begin to step onto the next, there is a sudden rise to the new level. Similarly, the greatest integer function exhibits a sharp increase at each new integer, leading to discontinuities.

For example, \( [2.9] = 2 \) and \( [3] = 3 \) demonstrate a jump from the value of 2 to 3 at the integer point. These jumps create gaps in the function's graph, which are precisely the locations of discontinuities. Therefore, we find that the greatest integer function is discontinuous at every integer value, as it abruptly shifts from one integer value to the next without taking on any value in between.

Fractional Part Function

Conversely, the fractional part function, denoted by \( (x) \) and defined as \( (x) = x - [x] \) captures the 'fractional' portion of \( x \), that is, what remains after subtracting the greatest integer less than or equal to \( x \). Unlike the greatest integer function, the graph of the fractional part function is characterized by a sawtooth pattern, rising linearly from 0 to just below 1 as \( x \) increases, then abruptly dropping back to 0 at the next integer value.

For instance, as \( x \) shifts from \( 2.9 \) to \( 3 \) the fractional part moves from \( 0.9 \) to \( 0 \), representing a clear discontinuity. The fractional part function resets to 0 at each integer, leading to a discontinuity at these points because the function does not approach the integer values smoothly, but rather resets abruptly.

Behavior of Mathematical Functions

Understanding the behavior of mathematical functions involves observing how these functions react or change as their input values vary. For the greatest integer and fractional part functions, their behaviors are quite distinct. Knowing these behaviors assists us in predicting function values and recognizing patterns, particularly when working with complex mathematical operations or solving problems.

The stair-step nature of the greatest integer function and the sawtooth oscillation of the fractional part function are unique signatures that help in identifying these functions on a graph. Awareness of these behaviors is vital, as it informs us about where to expect changes or constancy in function outputs. Particularly for these two functions, their predictable behavior at integer values makes them useful in a variety of mathematical contexts, though it also means they present a repetitive pattern of discontinuities.

Analyzing Discontinuities

Discontinuities occur in functions when there is a break, jump, or an abrupt change in the value of the function. In analyzing discontinuities, we're essentially seeking to pinpoint where a function does not have a 'connected' graph. For both the greatest integer and fractional part functions, we can observe these breaks at each integer value of \( x \).

The greatest integer function experiences a type of discontinuity known as a 'jump' discontinuity at each integer because it jumps from one value to the next. Similarly, the fractional part function also features jump discontinuities at integers due to the reset to 0. By recognizing these discontinuities, one can better understand function behaviors and the limits of certain mathematical operations. Moreover, discontinuities can have significant implications in the real world, particularly in scenarios that involve stepwise processes or cyclical patterns, much like the ones these functions represent.