Points of Discontinuity of Reciprocal Function

Identify the points of discontinuity for the function $f(x) = \frac{1}{x^2}$.

In solving this problem, we are tasked with identifying the points of discontinuity for the function given by one divided by the square of x. The primary concept to understand here is what makes a function discontinuous at a point. A function is discontinuous at a particular point if it is not defined there, or if it does not approach a specific finite value from both sides of the point.

For rational functions, such as one divided by x squared, discontinuities often occur at points where the denominator is zero, because division by zero is undefined. Here, our denominator is x squared. Hence, the function is not defined at x equals zero, leading us to conclude there is a discontinuity at this point. Importantly, this is a type of discontinuity known as an infinite discontinuity, because as x approaches zero, the function values increase or decrease without bound.

Understanding examples like this sharpens your insight into the broader topic of limits and continuity, a foundation for more advanced studies in calculus and real analysis. For deeper problem-solving, one should examine behavior on both sides of discontinuities, determining whether the limits from the left and right are finite or if they diverge, to further classify discontinuous points as removable, jump, or infinite discontinuities.