Question

Which one of the following statements is correct? $f(x) = \frac{1}{1 + \tan \, x}$

• A. is a continuous, real-valued function for all $x \in (- \infty,\infty)$
• B. is discontinuous only at $x = \frac{3\pi}{4}$
• C. has only finitely many discontinuities on $( - \infty, \infty)$
• D. has infinitely many discontinuities on $( - \infty, \infty)$

Answer 1

Answer: (D)

has infinitely many discontinuities on $( - \infty, \infty)$

Answer 2

tan x is not continuous at $x =\frac{\pi}{2}, 3 \frac{\pi}{2} , 5 \frac{\pi}{2}$etc... So, tan x has infinitely many discontinuities on $( - \infty ,\infty)$ $\Rightarrow \:\: f(x) = \frac{1}{1 + \tan \, x}$ has infinitely many discontinuities on $( - \infty ,\infty)$.