Continuity Rule
If the limit exists (aka left limit and right limit are equal), and the limit equals the function at that point.
The continuity rule, also known as the law of continuity, is a fundamental concept in mathematics that deals with the behavior of functions. It states that if a function f(x) is continuous over a closed interval [a, b] and g(x) is continuous over the closed interval [b, c], then the composite function (g ∘ f)(x), which represents performing g first and then f, is also continuous over the interval [a, c].
In simpler terms, if two functions are individually continuous on their respective intervals and are combined using composition, the resulting composite function will also be continuous on the larger interval.
To understand this concept visually, imagine a function f(x) that smoothly connects points on a graph from a to b and another function g(x) that smoothly connects points from b to c. If we combine these two functions by applying g to the output of f, the resulting composite function will also smoothly connect all the points from a to c. This seamless connection of points signifies continuity.
Mathematically, the continuity rule can be expressed as:
If f(x) is continuous at x = b and g(x) is continuous at y = f(b), then (g ∘ f)(x) is continuous at x = b.
This rule is essential in analyzing and proving the continuity of composite functions, particularly when combining multiple elementary functions to create more complex functions. It allows us to determine that combining continuous functions will always result in a continuous function.
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