Question

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 \(f(x)=3x-\cos x\) is continuous for all values of \(x\) ∈ Real Numbers. There are no discontinuous or removable discontinuities.

Step-by-step solution

Express the Function

Express the function: \(f(x)=3x-\cos x\). This is a combined function made of a linear function (3x) and a trigonometric function (cosx). Both of these functions are elementary and known to be continuous over Real numbers.

Determine Continuity of Functions

Determine the continuity of the individual functions, 3x and cosx are continuous for all x ∈ Real Numbers. The limit as x approaches any real number for these functions also exists.

:Evaluate the Overall Continuity

If a function can be expressed as the sum, difference, product or quotient of functions that are all continuous at a point, then the function itself is continuous at that point. Here, \(f(x)\) is the difference of two continuous functions. Hence, \(f(x)\) would be continuous for all Real x.

Identify Removable Discontinuities

In this case, there are no removable discontinuities since the function is continuous at every point in its domain.