Definition of Continuity
In mathematics, continuity is a fundamental concept that describes the behavior of functions
In mathematics, continuity is a fundamental concept that describes the behavior of functions. It is essential in understanding the smoothness and connectivity of functions on their entire domains.
A function is said to be continuous at a point if the graph of the function has no gaps, holes, or jumps at that particular point. Visually, this means that you can draw the graph of the function without lifting your pen from the paper.
Formally, a function f(x) is continuous at a point c if the following conditions are met:
1. The function is defined at c: This means that f(c) is well-defined and doesn’t have any undefined values or division by zero.
2. The limit of the function as x approaches c exists: This means that the left-hand limit (lim x->c- f(x)) and the right-hand limit (lim x->c+ f(x)) both exist and are equal to each other.
3. The limit of the function as x approaches c is equal to the value of the function at c: This means that lim x->c f(x) = f(c).
If all these conditions are satisfied, the function is said to be continuous at the point c.
Additionally, a function can be considered continuous on an interval (a, b) if it is continuous at every point within that interval. If a function is continuous for all real numbers, it is called a continuous function on the real line.
In summary, continuity is a property of functions that ensures the graph has a smooth, unbroken shape without any abrupt changes, and it is determined by the existence of a limit and its equality to the value of the function at a given point.
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