Problem 24 - Single Variable Calculus

Question

Find the numbers, if any, where the function is discontinuous. \(f(x)=\left\\{\begin{array}{ll}-|x|+1 & \text { if } x \neq 0 \\ 0 & \text { if } x=0\end{array}\right.\)

Step-by-Step Solution

Answer

The function \(f(x) = \left\{ \begin{array}{ll} -|x|+1 & \text { if } x \neq 0 \\ 0 & \text{ if } x=0 \end{array}\right.\) has no points of discontinuity. The left-hand limit and right-hand limit both equal 1 at \(x = 0\), and the function is continuous for \(x \neq 0\) since it involves an absolute value.

Step 1: Break down the piecewise function

The function is defined by two parts: 1. \(f(x) = -|x| + 1\) for \(x \neq 0\) 2. \(f(x) = 0\) for \(x = 0\)

Step 2: Analyze the intervals

The first part of the function is for all \(x\) except 0: 1. \(f(x) = -|x| + 1\) for \(x < 0\) and \(x > 0\). The second part of the function is for \(x = 0\): 2. \(f(x) = 0\).

Step 3: Check for discontinuity at x = 0

Check the left-hand limit (LHL) and the right-hand limit (RHL) of the function at \(x = 0\). \(LHL = \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (-|x| + 1)\) \(RHL = \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (-|x| + 1)\) Evaluate the limits: Since \(x < 0\), we have \(|x| = -x\) \(LHL = \lim_{x \to 0^-} (-(-x) + 1) = \lim_{x \to 0^-} (x + 1) = 1\) Since \(x > 0\), we have \(|x| = x\) \(RHL = \lim_{x \to 0^+} (-(x) + 1) = \lim_{x \to 0^+} (-x + 1) = 1\) Since LHL and RHL are equal, the function is continuous at \(x = 0\).

Step 4: Check for discontinuity for \(x ≠ 0\)

For \(x ≠ 0\), the function is given by \(f(x) = -|x| + 1\). This expression involves an absolute value, which is a continuous function. Since the function is continuous for \(x ≠ 0\), there is no other point of discontinuity. Thus, the function is continuous throughout its domain. There are no points of discontinuity.

Key Concepts

Piecewise Functions

Piecewise functions are particularly unique because they behave differently at various intervals of their domain. Think of them like a mathematical chameleon, changing their expression according to the values of the independent variable, which in this case is usually the variable 'x'.

Consider the function from our exercise, where we have one definition for all non-zero values of 'x', and another one specifically for when 'x' is zero. This allows the function to adapt itself to a particular context, much like how an equation for a real-life scenario might differ based on conditions or constraints.

Understanding piecewise functions involves identifying these distinct intervals and the corresponding definitions. Also, it requires careful examination of where these intervals meet, to determine the behavior of the function, especially at the borders of these intervals.

Limits and Continuity

The ideas of limits and continuity are central to understanding the behavior of functions, particularly around points where there's a potential for abrupt changes, such as with piecewise functions. A limit describes the value a function approaches as the input gets arbitrarily close to a particular point. Now, continuity is a sweet aftertaste of well-behaved limits: a function is continuous at a point if the limit at that point exists and equals the function's value there.

In the exercise, we checked for continuity at 'x = 0' by computing the left-hand limit (LHL) and the right-hand limit (RHL). If these two match, and if they also match the function's value at that point, the function is continuous there. Since our limits coincided and matched the function's value at 'x = 0', we declared it continuous without any interruption in its graph.

Absolute Value

The absolute value is the non-negative value of a number and is denoted by two vertical lines, like this: |x|. It represents the distance of a number from zero on the number line, regardless of the direction. For example, both 3 and -3 have an absolute value of 3.

In our function, the absolute value plays a pivotal role in ensuring the function behaves differently on either side of zero without actually breaking. However, despite its piecewise appearance due to the absolute value, the function maintains continuity because both sides approach the same value as 'x' approaches zero.

This continuous behavior through the absolute value function proves that it is indeed a cohesive glue holding the pieces of a piecewise function together, while transformingly sharply at our 'x = 0' checkpoint.