Question

Removable and Nonremovable Discontinuities In Exercises \(35-60,\) find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? \ $$ f(x)=3 x-\cos x $$

Short answer

The function \(f(x)=3x - \cos{x}\) has no discontinuities, so there are no removable or non-removable discontinuities.

Step-by-step solution

Analyze the components of the function

Firstly, we have to understand the components of the function \(f(x)=3x - \cos{x}\). This function is a combination of a linear function and a cosine function. Both of the components are continuous in their domains.

Identify the discontinuity

Secondly, we need to find any values of \(x\) where the function is not continuous. Since both the linear function and the cosine function are individually continuous for all real values of \(x\), their combination will also be continuous for all real \(x\). Hence, there are no discontinuities in this function.

Classify the discontinuity

Finally, we categorize any discontinuities as removable or non-removable. However, as we found out in the previous step, there is no discontinuity in \(f(x)=3x - \cos{x}\), so this step is not applicable in this case.

Key concepts

Removable Discontinuity

When we talk about discontinuities in a function, we're essentially discussing points at which a function isn't 'smooth' - it jumps, holes or breaks in some way. A removable discontinuity occurs at a certain point when a function has an 'interruption' at that point but can be 'fixed' by redefining the function at that point.

For example, consider the function defined by f(x) = (x^2 - 1) / (x - 1) for all x except x = 1. This function has a removable discontinuity at x = 1 because if you 'plug in' x = 1, you'd normally get a 0 in the denominator, which is undefined. But if you factor the numerator and cancel out the common terms, you get f(x) = x + 1, and the function becomes continuous if we define f(1) = 2.

In our original exercise, if there had been a discontinuity in f(x) = 3x - cos(x), and it was of such a nature that we could redefine the function to be continuous, then it would be a removable discontinuity. However, no such points exist for this specific function.

Nonremovable Discontinuity

Unlike a removable discontinuity, a nonremovable discontinuity is one that cannot be 'fixed' by simply redefining the function at a point of discontinuity. There are several types, with the most common being jump and infinite discontinuities.

Jump discontinuities occur when a function has a 'jump' between two values as x approaches a particular point from the left and the right. This often happens in piecewise functions where different expressions define different intervals. Infinite discontinuities happen when the function approaches infinity at a certain point along the x-axis.

If the function f(x) = 3x - cos(x) from our exercise had possessed a discontinuity, and it was not possible to make the function continuous by redefining the function at that point, then we would classify this point as a nonremovable discontinuity. In practice, for this function, that situation does not occur as it is continuous for all real numbers.

Continuity in Calculus

The concept of continuity is foundational in calculus. It describes a function that is smooth, unbroken, and predictable in behavior over its domain. In formal terms, a function f(x) is continuous at a point x = a if three conditions are met:

• The function is defined at a; that is, f(a) exists.
• The limit of f(x) as x approaches a exists.
• The limit of f(x) as x approaches a is equal to f(a).

Continuity checks whether you can draw the graph of a function without lifting your pencil off the paper. For the function in our example, f(x) = 3x - cos(x), this condition holds true for all x in its domain, reflecting its smooth nature without any jumps, breaks, or holes.

Cosine Function

The cosine function, denoted by cos(x), is a fundamental trigonometric function that arises throughout mathematics. Particularly in calculus, the cosine function is a periodic function meaning it repeats its values in a predictable pattern called a cycle. The cos(x) function is continuous over all real numbers, which implies there are no breaks or jumps in its graph.

It is defined for any angle in a circle and is particularly known for its relationship to the x-coordinate of a point on a unit circle. A unit circle is a circle with a radius of one, centered at the origin of a two-dimensional coordinate plane. In the context of the function f(x) = 3x - cos(x), the - cos(x) portion contributes a smooth wave-like component to the linear term, resulting in a continuous function that oscillates around the line y = 3x, without any discontinuities.