Understanding the Concept of Continuity in Mathematics | Explained and Illustrated.

Definition of Continuity

In mathematics, continuity refers to the property of a function or a curve that it has no abrupt changes or disruptions

In mathematics, continuity refers to the property of a function or a curve that it has no abrupt changes or disruptions. It implies that the function or curve can be freely drawn without lifting the pen or encountering any holes, breaks, or jumps.

More formally, a function f(x) is said to be continuous at a point x = c if three conditions are met:
1. The function is defined at x = c.
2. The limit of the function as it approaches c exists.
3. The value of the function at x = c is equal to the limit mentioned above.

Mathematically, these conditions can be written as:
1. f(c) is defined.
2. The limit of f(x) as x approaches c, denoted by lim (x→c) f(x), exists.
3. lim (x→c) f(x) = f(c).

If these conditions are satisfied for every point c in the domain of the function, then the function is considered continuous on its entire domain.

Furthermore, continuity can also be extended to intervals. A function is continuous on an interval if it is continuous at every point within that interval.

Continuity is a fundamental concept in calculus and analysis as it allows for the study and analysis of functions and their behavior without any interruptions or abrupt changes.

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