Math Help

Conditions for Continuity of a Function, Continuous Functions

In a Nut Shell: In the discussion of limits of a function, the notion of a “smooth”

curve was used. It relates directly to the notion of continuity of a function.

The definition of continuity of a function is as follows:

f(x) is said to be continuous at x = a (in a neighborhood of a) provided that

lim f(x) = f(a)

x → a

Three conditions must exist for continuity of a function. They are:

· the function must be defined at x = a

· the limit of f(x) as x → a must exist

· lim f(x) = f(a)

x → a

If any of these conditions is not satisfied, then f(x) is discontinuous at x = a.

Common continuous functions include:

· polynomial functions are continuous on x

· rational functions , p(x)/q(x) are continuous everywhere on x provided that

q(x) ≠ 0. The discontinuity where q(x) = 0 might be removable.

· trigonometric functions sin x, cos x are continuous on x

· the root of a continuous function is continuous wherever it is defined

Squeeze Law Suppose that f(x) ≤ g(x) ≤ h(x) for all x ≠ a in some neighborhood

of a and also that

lim f(x) = L = lim h(x)

x → a x → a

then lim g(x) = L

x → a

Hint: Suppose g(x) is a complicated function and you can find two simpler functions, f(x) and

h(x) that bound g(x). Then you can use limits on f(x) and h(x) to find the limit on g(x).