Problem 48 - A Textbook of Differential Calculus

Question

\(\operatorname{Max}([x],|x|)\) is discontinuous at, (a) \(x=0\) (b) \(\phi\) (c) \(x=n, n \in I\) (d) None of these

Step-by-Step Solution

Answer

(c) \(x=n, n \in \mathbb{I}\)

Step 1: Understand the Function Components

The function in question is \(\operatorname{Max}([x], |x|)\), which means it selects the maximum of the floor value \([x]\) and the absolute value \(|x|\). The floor function \([x]\) represents the largest integer less than or equal to \(x\), and \(|x|\) is the absolute value of \(x\).

Step 2: Investigate Discontinuities in the Floor Function

The floor function \([x]\) is discontinuous at every integer \(n\). For \(x = n + \epsilon\) (where \(0 < \epsilon < 1\)), \([x] = n\), but for \(x = n^-\), \([x] = n - 1\). This jump by 1 at every integer causes a discontinuity.

Step 3: Examine the Absolute Value Function

The absolute value function \(|x|\) is continuous for all real numbers as it smoothly transforms \(x\) into its non-negative equivalent without any jumps or breaks.

Step 4: Analyze Decremnt \(\operatorname{Max}([x], |x|)\) for Discontinuity

Discontinuity in \(\operatorname{Max}([x], |x|)\) could arise if \([x]\) jumps above \(|x|\) for an interval close to integers \(n\), causing the output to switch abruptly. This would occur at these integer points where \(\operatorname{Max}([x], |x|)\) changes its value.

Step 5: Conclusion on Discontinuity of \(\operatorname{Max}([x], |x|)\)

The function \(\operatorname{Max}([x], |x|)\) is discontinuous at every integer \(n\), because the sudden jump in the floor component at integers can override another smoothly continuous absolute value function, creating a discrete change in the resulting function value.

Key Concepts

Floor Function

In mathematics, the floor function, denoted as \([x]\), is used to round down a real number to the nearest integer less or equal to that number. It means, if you have a number like 3.7, its floor value is 3. Likewise, the floor of -2.3 is -3 because -3 is the closest integer that is less than -2.3.
Working with the floor function involves understanding this fundamental rule:

• If \(x\) is an integer, \([x] = x\).
• If \(x\) is not an integer, \([x]\) jumps down to the next lower integer.

At every integer point, \([x]\) is not just a simple discrete integer. When approaching integer values from below, the function immediately drops to that integer, and this creates what’s called a 'jump discontinuity' or 'step' in the graph of the function, which occurs exactly at every integer value.

Absolute Value Function

The absolute value function, represented as \(|x|\), is a simple yet foundational concept in mathematics that measures the distance between a number and zero on the number line. It essentially converts every number to its non-negative form.
To compute the absolute value:

• If \(x\) is a positive number, \(|x| = x\).
• If \(x\) is a negative number, \(|x| = -x\).
• If \(x\) is zero, \(|x| = 0\).

Importantly, this function is continuous for all real numbers. Unlike the floor function, the absolute value does not create any sharp changes or jumps, allowing the function \(|x|\) to have a smooth line when graphed.

Discontinuous Points

A function is said to be discontinuous at a certain point if there is an abrupt change in its behavior. Such points are where the function literally "jumps" from one value to another with no smooth transition. This usually happens when there's a sudden leap, increase, or decrease at a spot.
In the context of \(\operatorname{Max}([x], |x|)\), discontinuity arises specifically when the behavior of \([x]\) and \(|x|\) compete. If \([x]\) makes a sudden jump at an integer, it might override \(|x|\), which is smooth and steady, causing a shift in the output value of the max function at that point.

Integer Discontinuity

Integer discontinuity in functions occurs primarily because of functions like \([x]\), the floor function. Approaching an integer \(n\), the floor value drops abruptly to \(n\), creating an integer discontinuity. Such points of discontinuity are sharp, causing the overall function to experience an obvious jump at every whole number.
For \(\operatorname{Max}([x], |x|)\), as you get to an integer \(n\), the value of \([x]\) can suddenly increase or decrease in comparison to \(|x|\). This affects which part of \(\operatorname{Max}([x], |x|)\) is dominant. At these spots, where the floor value leaps more significantly than the absolute value alters, the dominant \([x]\) imposes a break in the function's smoothness, leading to integer-driven discontinuities.