What are the kinds of discontinuity?

The correct option is First and second kinds.

In mathematical analysis and calculus, the discontinuities of a real-valued function are classically categorized into two primary classifications: discontinuities of the first kind and discontinuities of the second kind.

Let us break down these two classifications to understand why this terminology is used:
Consider a function f(x) defined on an interval around a point c (except possibly at c itself). The classification of discontinuity at x = c depends on the existence of the one-sided limits:

The left-hand limit is defined as:

$$\underset{x\to c^{-}}{lim}f\left(x\right)$$

The right-hand limit is defined as:

$$\underset{x\to c^{+}}{lim}f\left(x\right)$$

1. Discontinuity of the First Kind:
A discontinuity at x = c is said to be of the first kind if both the left-hand limit and the right-hand limit exist as finite numbers. Within this kind, there are two subcategories:
• Removable Discontinuity: The left-hand and right-hand limits exist and are equal, but they do not equal the value of the function at that point, i.e., f(c).
• Jump Discontinuity: The left-hand and right-hand limits exist but are not equal to each other.

2. Discontinuity of the Second Kind (Essential Discontinuity):
A discontinuity at x = c is said to be of the second kind if at least one of the one-sided limits (either left-hand or right-hand) does not exist as a finite limit (for instance, if the function oscillates infinitely or goes to positive/negative infinity as it approaches c).

Thus, the standard classification divides discontinuities into the first and second kinds.