Definition of Continuity
Continuity is a fundamental concept in mathematics that describes the behavior of a function
Continuity is a fundamental concept in mathematics that describes the behavior of a function. It is a property that determines whether a function can be drawn without lifting your pencil from the paper.
A function f(x) is said to be continuous at a point x = a if three conditions are met:
1. The function is defined at that point, meaning f(a) is well-defined.
2. The limit of the function as x approaches a exists, denoted by limₓ→a f(x), and it is equal to f(a).
3. The limit of the function as x approaches a exists from both the left and the right, and the left and right limits are equal. Mathematically, we express this as: limₓ→a⁺ f(x) = limₓ→a⁻ f(x) = f(a), where ⁺ denotes approaching from the right and ⁻ denotes approaching from the left.
In simpler terms, a function is continuous at a point if its graph has no holes, jumps, or any other kind of break. This means that as you approach that point from either side, the function gets arbitrarily close to the same value as the point itself.
We can also extend this definition to state that a function is continuous on an interval if it is continuous at every point within that interval.
The concept of continuity is essential in calculus and analysis, as it allows us to analyze the behavior of functions and apply various mathematical techniques. It is also a key building block in understanding limits, derivatives, and integrals. By studying the continuity of a function, we can make predictions about its behavior and analyze its properties.
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