What is a discontinuous function? Discuss the two types of discontinuous functions along with their diagrams

A discontinuous function is a mathematical function that does not have continuity at one or more points in its domain.

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Continuity is a property that implies the function has no abrupt jumps, holes, or vertical asymptotes.

There are two main types of discontinuous functions: jump discontinuity and essential discontinuity.

1. Jump Discontinuity:

• Definition: A function has a jump discontinuity at a point if the left-hand limit and right-hand limit exist at that point, but the function value “jumps” from one side to the other.
• Diagram:
  f(x) | * | * | * | * Jump discontinuity at x = a |______*_____ a

1. Essential Discontinuity:

• Definition: A function has an essential discontinuity at a point if at least one of the one-sided limits (left-hand or right-hand limit) does not exist, or both one-sided limits exist but are unequal.
• Diagram:
  f(x) | * | * | * | * Essential discontinuity at x = a |__*________ a

These diagrams provide a visual representation of the two types of discontinuities. In a jump discontinuity, the function value jumps from one level to another, creating a visible gap. In an essential discontinuity, there is typically a hole or an asymptote, indicating a more complex behavior at that point.

Understanding these types of discontinuities is crucial in analyzing the behavior of functions, especially when dealing with limits and the continuity of functions at specific points.