The number of points at which the function $f(x) =\frac{1}{\log|x|}$ is discontinuous is

3

The function log |x| is not defined at x = 0, and hence x = 0 is a point of discontinuity.

Also, for f(x) to be defined, log|x| ≠ 0 that is x ≠ ±1.

Hence, 1 and –1 are also points of discontinuity.

Clearly f(x) is continuous for x ∈ R – {0, 1, –1}.

Thus, there are three points of discontinuity.

Hence (C) is the correct answer.