CONTINUITY

Definition: A function f is continuous at a point x = a if

In other words, the function f is continuous at a if ALL three of the

conditions below are true:

1. f (a) is defined.

(i.e., a is in the domain of f .)

2. exists.

(i.e., both one-sided limits exist and are equal at a.)

3.

(i.e., the results of conditions 1 and 2 are the same.)

If any one of the conditions is false, then we say that f is discontinuous at a, or that it has a discontinuity at a.

A consequence of this definition is that if we know a function f is

continuous at a point x = a, then we also know that it has a limit at a, equal to f (a).

Consider the following examples and determine if the function is continuous at x = 4.

1. . Is this function continuous at x = 4?

2. g(x) = for AND g(x) = 5 for x = 4.

3. h(x) = for AND h(x) = 8 for x = 4.

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Types of discontinuities:

· Removable discontinuity

· Infinite discontinuity

· Jump discontinuity

How to identify the type of a discontinuity?

· Removable Discontinuity at x = a we could define a new nearly identical function which would fill the discontinuity at x = a. (See above examples.) This occurs when the second continuity condition is satisfied (i.e. exists), but either the first or third condition fails.

· Infinite Discontinuity at x = a if one or both of the one-sided limits is or .

· Jump Discontinuity at x = a if both of the one-sided limits exist, but they are not equal.

More examples – Determine any values of x for which the function is discontinuous. Graph the functions and determine the type of discontinuity.

4. .

5.

6.

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We need to be especially concerned with potential discontinuities in piecewise functions where the domain is divided. For example, check for continuity at x = 0 and 2.

7. for , for , for x >2.

LEFT AND RIGHT CONTINUITY

A function f is said to be continuous from the left at x = a if

A function f is said to be continuous from the right at x = a if

If f(x) is both left continuous and right continuous at x = a, then it is continuous at x = a.

8. Consider example 7 above. Determine left and right continuity at x = 0 and x = 2.

A function f is said to be continuous on an interval if it is continuous at each and every point in the interval. A function may only be left or right continuous at an endpoint.

Theorem: (4, pg. 100)

If f and g are continuous at a and c is a constant, then the following functions are also continuous at a.

1. f + g 2. f – g 3. cf 4. f g 5. f/g (provided )

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Theorem: (5, pg. 100)

(i.) Every polynomial function is continuous everywhere on .

(ii.) Every rational function is continuous everywhere it is defined, i.e., at every point in its domain. Its only discontinuities occur at the zeros of its denominator.

9. is continuous on .

10. is continuous on .

11. Consider combinations of function as given in Theorem 4.