Question

Find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? $$ f(x)=\tan \frac{\pi x}{4} $$

Short answer

Function \(f(x) = \tan(\frac{\pi x}{4})\) is not continuous at \(x = 4n\) where \(n\) is any odd integer and none of these discontinuities are removable.

Step-by-step solution

Identify Discontinuities

As the denominator of the tangent function is cosine, and \(\cos(x)\) isn't defined at odd multiples of \(\frac{\pi}{2}\), so we set the argument \(\frac{\pi x}{4}\) equal to odd multiples of \(\frac{\pi}{2}\) i.e. \(n\pi\), where \(n\) is odd. Solving this equation \(\frac{\pi x}{4} = n\pi\) where \(n\) is odd gives x-values.

Find X-values

Solving the equation \(\frac{\pi x}{4} = n\pi\) for \(x\) where \(n\) is odd gives us \(x = 4n\). So, for all odd values of n, function \(f(x)\) will not be continuous at \(x = 4n\). Thus function has discontinuities at \(x = 4n\) where \(n\) is any odd integer.

Determine Removable Discontinuities

A discontinuity is removable if the function can be redefined in a way that it can become continuous. The given function cannot be redefined into a continuous function by removing or adding points, because the tangent function has infinite discontinuities at odd multiples of \(\frac{\pi}{2}\). Therefore none of the discontinuities is removable in this case.