Problem 46 - Calculus Early Transcendentals: Pearson New International Edition

Question

Determine whether the function is continuous at the given point \(c .\) If the function is not continuous, determine whether the discontinuity is removable or non-removable. $$ F(x)=x \sin \frac{1}{x} ; c=0 $$

Step-by-Step Solution

Answer

Removable discontinuity.

Step 1: Check continuity conditions

A function is continuous at \(c\) if: (1) \(f(c)\) is defined, (2) \(\lim_{x\to c} f(x)\) exists, and (3) \(\lim_{x\to c} f(x) = f(c)\).

Step 2: Classify the discontinuity

If the limit exists but doesn't equal \(f(c)\) (or \(f(c)\) is undefined), the discontinuity is removable. If the limit doesn't exist (e.g., jump or infinite), it is non-removable.

Key Concepts

Continuity

In calculus, a function is said to be continuous at a particular point if there are no interruptions or jumps in the values of the function around that point. For a function to be continuous at a point \( c \), three conditions must be satisfied:

• The function \( f(x) \) must be defined at the point \( c \).
• The limit of \( f(x) \) as \( x \) approaches \( c \) must exist.
• The limit of \( f(x) \) as \( x \) approaches \( c \) must be equal to \( f(c) \).

When these conditions are met, the graph of the function is smooth at \( c \), without any breaks or holes. This means you can draw the function's graph at that point without lifting your pencil. It's useful to remember that while all polynomial functions are continuous everywhere, functions with division or other operations might have points where continuity fails.

Discontinuity

If a function does not satisfy the conditions for continuity at a specific point, it is said to have a discontinuity at that point. Discontinuities can take various forms, including jumps, infinite behavior, or oscillations. For the function \( F(x) = x \sin \frac{1}{x} \), we are interested in its behavior as \( x \) approaches 0.

This particular type of function combines both algebraic and trigonometric elements, which can lead to complex behavior near zero. Identifying whether a function has a discontinuity involves checking the limit and ensuring there are no values where the function becomes undefined. At points of discontinuity, the graph of the function might show gaps, jumps, or holes – signals that the smooth progression of the graph is interrupted.

Removable Discontinuity

A removable discontinuity occurs at a point where a function is not continuous, but the limit of the function exists, and by redefining the function at that point, continuity can be restored. These are often due to factors such as division by zero or undefined points that lead to holes in the graph.

• When you graph the function, a removable discontinuity appears as a hole at a specific point.
• Graphically, you can "fill in" this hole by redefining the function value at this point, thus restoring continuity.

In the given problem with \( F(x) = x \sin \frac{1}{x} \) at \( c = 0 \), we consider whether the limit exists as \( x \) approaches 0. If the limit can be computed and is finite, the discontinuity may be removable. Redefining \( F(0) \) to this limit value would make the function continuous at \( c = 0 \). This process converts a removable discontinuity into a continuous point.

Non-removable Discontinuity

Non-removable discontinuities are those that cannot be "fixed" by redefining the function at the point of discontinuity. This type of discontinuity usually involves the behaviors like jumps or infinite limits.

• In cases of jump discontinuity, the function value leaps from one point to another with no intermediate values.
• In infinite discontinuities, the function approaches infinity, one way or another, as \( x \) approaches the point.

For these, no matter how you define the value at the point, the function remains discontinuous. In the context of \( F(x) = x \sin \frac{1}{x} \) at \( c = 0 \), if the behavior of the function as it approaches 0 does not settle to a single value or becomes infinite, the discontinuity is non-removable. Here, we would look for indicators such as a lack of clear limit or undefined behavior to classify it as non-removable.