Discontinuous at a point

Definition

A function is discontinuous at a point if there is a sudden jump, break, or hole at that point in its graph. The function does not have a well-defined limit or the value of the function does not match the limit at that point.

5 Must Know Facts For Your Next Test

1. For a function f(x) to be continuous at x = c, $\lim_{{x \to c}} f(x)$ must exist and be equal to f(c).
2. There are three types of discontinuities: removable, jump, and infinite.
3. Removable discontinuity occurs when $\lim_{{x \to c}} f(x)$ exists but is not equal to f(c).
4. Jump discontinuity happens when $\lim_{{x \to c^-}} f(x) \neq \lim_{{x \to c^+}} f(x)$. The left-hand limit does not equal the right-hand limit.
5. Infinite discontinuity arises when the limits approach infinity as x approaches c from either side.

Review Questions

• What conditions must hold for a function to be continuous at a point?
• Explain the difference between removable and jump discontinuities.
• How do you identify an infinite discontinuity on a graph?

Related terms

Limit: The value that a function approaches as its input approaches some value.

Continuity: A property of functions where they are continuous at every point in their domain.

Removable Discontinuity: $\lim_{{x \to c}} f(x)$ exists but is not equal to f(c), often indicating a hole in the graph.