B.6) Continuous functions

B.6.a) Definitions

The really nice examples of limits are the ones where there were no difficuties to overcome--no holes in graphs, no jumps, no wild wiggles. These are functions where limits give exactly the expected result. Most functions we encounter in the real world are well behaved in this way, so it is worth giving a name:

Definition: A function f is continuous at a if f[x]= f[a].

This definition says three things:
1) The limit of f[x] as x approaches a must exist.
2) The function must have a value at a.
3) These two values must be the same.

Wild bouncing or jumps will cause a function not to have a limit, violating the first of these; a hole in the graph would violate the second; and a misplaced value would violate the third. Vertical asymptotes happen where limits give ± , which we take for this definition as not having a limit. So a function will be discontinuous at any vertical asymptote.

We are usually interested in functions that are continuous at every point in a given set. Here are two basic definitions.

Definition: A function f is continuous on an open interval (a, b) if it is continuous at every point in that interval. It is continuous on a closed interval [a, b] if in addition the value of the function at each endpoint agrees with the value of the one-sided limit as you approach that endpoint.

B.6.b) Examples

As mentioned before, most of the functions that occur in the study of calculus are continuous at every point in their domains. The major problems occur at places where we are either dividing by 0 or evaluating Log[0]. Either of these things can lead to holes, vertical asymptotes, or wild wiggles in the graph. Since the function is not defined at these touble spots, they occur outside of the natural domain of the function. Here's a list of common functions, which are continous on their domains, and their domains:
lines and polynomials --- domain = (-, )
rational functions --- domain = all reals where the bottom
is not equal to 0
exponential functions --- domain = (-, )
Logs --- domain = (0, )
Sine and Cosine --- domain = (-, )
Tangent and Secant --- domain = all reals except
x = /2 + k where k is any integer (i.e. except
where Cos[x] = 0).
Cotangent and Cosecant --- domain = all reals except
x = k where k is any integer (i.e. except where
Sin[x] = 0).
Power functions with non-integer exponents --- domain
varies:
x^r where r is irrational --- domain = (0, )
x^(p/q) with p, q integers and q is even --- domain
= (0, )
x^(p/q) with p, q are integers and q is odd --- domain
= (-, )