Calculus: Continuity

A definition for "continuous" from Dictionary.com is "being in immediate connection or spatial relationship." Mathematical continuity is the same, roughly speaking. A function f(x) is continuous at a value a if the limit of the function as x approaches a is equal to the functional value at a. In other words: 

This simple definition actually contains three implications.

Many functions are continuous over their entire domains, such as polynomials and rational functions. The domain of a polynomial function is all real numbers and the domain of rational functions are everywhere except at vertical asymptotes. Thus, to find the limit of a polynomial or rational functions (that are not restricted in any way) at a particular value, one must simply find the functional value at the desired x-value. Easy enough, right?

A function can be discontinuous at a value a if the above requirements are not met. There are multiple types of discontinuity.

A function can be continuous from only one side and we call that left-hand or right-hand continuity. A function is continuous over an interval if the function is continuous at every value in the interval. You can use the definition of continuity (using limits) to determine if a function is continuous over an interval or at a value.

You can also use the properties of continuity to determine if a function is continuous. Let f and g be continuous functions at a and let c be a constant. Then the following functions are also continuous at a:

1. f + g
2. f – g
3. cf
4. fg
5. f/g, provided g(a) does not equal zero

If you are asked to determine if a function is continuous at a value a, first see if you can break it into two functions (f and g) that you know are continuous, probably some combination of rational and polynomial functions. Then the above properties tell you that these combinations of two continuous functions are also continuous.

Many functions are continuous over their entire domains, not just polynomials and rationals. Here is a list: polynomials, rational, root, trigonometric, inverse trigonometric, exponential, and logarithmic functions are all continuous at every number in their domains. Thus, any sum or difference (properties 1 and 2) of these, any multiple of these (3), any product of these (4), and any quotient of these (5; provided the denominator is not zero) is also continuous at a, provided that a is in the domain of both continuous functions.

If g is continuous at a and f is continuous at g(a), then the composite function f(g(x)) is also continuous at a. Note that here, f(x) does not need to be continuous at a, but at g(a) because g(a) is what is being "fed" into the f(x) function in the case of composite functions.

Intermediate Value Theorem: Suppose f is continuous on the closed interval [a, b] and let N be any number between f(a) and f(b), where f(a) does not equal f(b). Then there must exist a number c in (a, b) such that f(c) = N.

This sounds confusing, but it makes perfect sense. If a function is continuous over an interval, then it is continuous at every number in that interval. If there is a y-value, N, between two other y-values, f(a) and f(b), which are all in this continuous interval, then there must be some x-value whose functional value is N, and we call that value c. This value occurs in the open interval (a, b). Since the function is continuous over this interval, we can think of it as not being "broken" between these x-values a and b. Since it's not broken, every x-value in this interval has a y-value and vice-versa: every y-value in the interval (f(a), f(b)) must have an x-value in the interval (a, b).

We will use continuity in further Calculus topics and the Intermediate Value Theorem (IVT) is used in many Calculus proofs, so you should become familiar with it and work some problems involving it before you move to the next topic of interest.