We have seen that any polynomial function P(x) satisfies:
Definition. Let f(x) be a function defined on an interval
around a. We say that f(x) is continuous at a iff
Note that the continuity of f(x) at a means two things:
Basic properties of limits imply the following:
Theorem. If f(x) and g(x) are continuous at a. Then
Remark. Many functions are not defined on open intervals.
In this case, we can talk about one-sided continuity. Indeed,
f(x) is said to be continuous from the left at a iff
Example. The function
is defined for .
So we can not talk about left-continuity of f(x) at
0. But since
This concept is also important for step-functions.
Example. Consider the function
Exercise 1. Find A which makes the function
Definition. For a function f(x) defined on a set S, we say that f(x) is continuous on S iff f(x) is continuous for all .
Example. We have seen that polynomial functions are continuous on the entire set of real numbers. The same result holds for the trigonometric functions and .
The following two exercises discuss a type of functions hard to visualize. But still one can study their continuity properties.
Exercise 2. Discuss the continuity of
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Mohamed A. Khamsi