Question

Testing for Continuity In Exercises \(77-84\) , describe the interval(s) on which the function is continuous. $$ f(x)=\cos \frac{1}{x} $$

Short answer

The function \(f(x)=\cos \frac{1}{x}\) is continuous for all real number x except for x=0.

Step-by-step solution

Identify the Critical Point

The critical point for the function exists where the function may become undefined or where the denominator becomes zero. For the function \(f(x)=\cos \frac{1}{x}\), the denominator, x, should never be zero. Therefore, x=0 is a critical point.

Determine the Intervals of Continuity

Except for the critical point at x=0, the function \(f(x)=\cos \frac{1}{x}\) is continuous everywhere else. Thus, the intervals of continuity for the function are (-infinity, 0) and (0, +infinity).

Combine the Intervals of Continuity

The intervals can be combined into one as (-infinity, 0) union (0, +infinity), meaning the function is continuous for all x not equal to 0.

Key concepts

Critical Point

When delving into the realm of calculus, the concept of a critical point is pivotal. A critical point of a function is where its derivative is zero or undefined, indicating a potential maximum, minimum, or inflection point.

From the exercise, we identify the critical point by looking at where the function may become undefined. In the case of the function \(f(x)=\text{cos}\left(\frac{1}{x}\right)\), we must consider the behavior as x approaches zero. Since cos(1/x) approaches infinity when x tends to zero, the function is undefined there, thus making \(x=0\) a critical point.

Understanding critical points is essential because they help map the terrain of a function's graph. In practical terms, knowing where these points are located can lead us to important conclusions about the function's behavior—such as identifying peaks in a profit function or lowest stress points in a physical structure.

Intervals of Continuity

The intervals of continuity are segments within the domain of a function where it does not experience any breaks or interruptions. To pinpoint these intervals, one must inspect the function across its entire domain and look for any discontinuities.

For the function \(f(x)=\text{cos}\left(\frac{1}{x}\right)\), except for the critical point at \(x=0\), the function is actually continuous. Hence, the intervals of continuity extend from negative to positive infinity but do not include the point where x equals zero. More formally, these intervals are written as \((-\infty, 0) \cup (0, +\infty)\), signifying that the function is steady and uninterrupted for all real numbers except zero.

Students should remember that continuity implies a smooth path — if you were to draw the graph of a continuous function, you would not need to lift your pencil off the paper within these intervals. Checking for continuity across a function's domain is comparable to ensuring there are no gaps in a bridge.

Undefined Functions

Certain functions encounter points or intervals where they are undefined. These are often places where an operation within the function's formula cannot be completed, like division by zero or taking the logarithm of a negative number.

In the given exercise, the function \(f(x)=\text{cos}\left(\frac{1}{x}\right)\) becomes undefined when x equals zero because it poses a situation of division by zero, which is an operation without meaning in conventional mathematics. Such undefined points create disruptions in the graph of the function, as the value of the function cannot be determined at these points.

It is crucial for students to learn to spot when a function is undefined to avoid misconceptions and incorrect calculations. Whether drafting graphs, solving equations, or simply working through a function's properties, recognizing undefined points prevents the oversight of potential discontinuities and ensures a thorough understanding of a function's behavior.