What is discontinuous in math?

As a mathematician with a deep understanding of the subject, I would like to shed light on the concept of discontinuity in mathematics. Discontinuity refers to a point or a set of points where a function or a sequence fails to be continuous. To grasp this concept, it's essential to understand what continuity means first. A function is said to be continuous if, roughly speaking, it is unbroken and has no gaps or jumps. This means that for any point in the domain of the function, there is a smooth transition to the corresponding point in the codomain. However, when a function is discontinuous, it exhibits a break or a jump at certain points, which can be classified into different types based on the nature of the break. ### Types of Discontinuities 1. Removable Discontinuity: This is the simplest type of discontinuity. It occurs when the function is undefined at a point, but the function can be redefined at that point to make it continuous. For instance, a function like \( f(x) = \frac{x^2 - 1}{x - 1} \) has a removable discontinuity at \( x = 1 \) because it can be simplified to \( f(x) = x + 1 \) for all \( x \) except \( x = 1 \). If we redefine \( f(1) \) to be 2, the discontinuity is removed. 2. Infinite Discontinuity: A function has an infinite discontinuity at a point if the function's value at that point is undefined because it tends towards infinity. An example is \( g(x) = \frac{1}{x} \), which has an infinite discontinuity at \( x = 0 \) because as \( x \) approaches 0, \( g(x) \) tends towards infinity. 3. Jump Discontinuity: This type of discontinuity occurs when the left and right limits of the function at a point exist but are not equal. The function "jumps" from one value to another without a smooth transition. A classic example is the absolute value function \( h(x) = |x| \), which has a jump discontinuity at \( x = 0 \) because the left and right limits are \( -1 \) and \( 1 \), respectively. 4. Oscillatory Discontinuity: Also known as non-removable discontinuity, this occurs when the function oscillates between different values in an interval around a point. A common example is \( k(x) = \sin\left(\frac{1}{x}\right) \), which has an oscillatory discontinuity at \( x = 0 \) because the function oscillates infinitely as \( x \) approaches 0. 5. Step Discontinuity: This is a specific type of jump discontinuity where the function changes by a fixed amount at a point. It's called a step because the graph of the function resembles a step on a staircase. ### Graphical Representation When visualizing discontinuities, one-variable functions typically display a break in the graph, while for two-variable functions, the discontinuity might be represented as a "tear" or "gap" in the surface plot. ### Mathematical Rigor Mathematically, a function \( f \) is continuous at a point \( c \) in its domain if the following three conditions are met: - \( f(c) \) is defined. - The limit of \( f(x) \) as \( x \) approaches \( c \) exists. - The limit of \( f(x) \) as \( x \) approaches \( c \) is equal to \( f(c) \). If any of these conditions fail, the function is said to be discontinuous at that point. ### Conclusion Understanding discontinuities is crucial in various fields of mathematics and its applications, from calculus to engineering. They help us identify points where functions behave unexpectedly and can affect the behavior of mathematical models. Recognizing and classifying discontinuities is key to solving problems and making accurate predictions in these areas.

A discontinuity is point at which a mathematical object is discontinuous. The left figure above illustrates a discontinuity in a one-variable function while the right figure illustrates a discontinuity of a two-variable function plotted as a surface in . ... The simplest type is the so-called removable discontinuity.