## Continuity

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# Continuity

Many functions have the property that there is no “break” in their graphs. For example, consider the graph of the function f as shown in. When x = a, the graph of the function of is unbroken.

Stated another way, if we were to trace the graph of f, we would not have to lift the pencil form the paper. However, if we consider the graph of the function g as shown in, we see that it has a break at x = a. In other words, if we were to trace its graph, we would have to lift the pencil on eh graph of g when x = . We char-acterize these situations by saying that f is continuous at x = a and g is discontinuous at x = a.

However, we must clearity the concept of continuity and make it more precise without a reference to the graph of a function.

Definition. (Continuous Function). A function y = f (x) is said to be continuous at x = a if

(1) f (a) exists, i.e.f is defined at x = a.

(2) lim f (x) exists.

x→a

(3) lim f (x) = f (a), i.e., limit of function equals function value.

x→a

A function is said to be continuous over an interval if the function is continuous at each point of the interval. A function which is into continuous at a point ‘a’ is said to be discontinuous at a.

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Stated another way, if we were to trace the graph of f, we would not have to lift the pencil form the paper. However, if we consider the graph of the function g as shown in, we see that it has a break at x = a. In other words, if we were to trace its graph, we would have to lift the pencil on eh graph of g when x = . We char-acterize these situations by saying that f is continuous at x = a and g is discontinuous at x = a.

However, we must clearity the concept of continuity and make it more precise without a reference to the graph of a function.

Definition. (Continuous Function). A function y = f (x) is said to be continuous at x = a if

(1) f (a) exists, i.e.f is defined at x = a.

(2) lim f (x) exists.

x→a

(3) lim f (x) = f (a), i.e., limit of function equals function value.

x→a

A function is said to be continuous over an interval if the function is continuous at each point of the interval. A function which is into continuous at a point ‘a’ is said to be discontinuous at a.

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