Question

Suppose f(x) = [sin(x^2 - 4)]^ -1. Identify any points of discontinuity, and determine (giving reasons) if they are removable, infinite (essential), or jump discontinuities.

Okay, I presume that the [] brackets denote the greatest integer function (int () ). Once I graphed the function on my graphing calculator, it returned a tragically ugly line of dots along y = -1. How can I interpret this and describe it for every single point?

Answer 1

Actually, in spite of what I wrote in an earlier post, it's possible that the brackets are used just to avoid having nested parentheses. In that case, f(x) has discontinuities wherever sin(x^2-4) is zero.

That is, where x^2 - 4 is a multiple of pi.

If x^2 - 4 = k*pi,

Then f is discontinuous at x = sqrt(k*pi + 4) for all integer k