Summary

Continuity A function f is continuous at x=c if the following three conditions are met:

1. f(c) is defined,
2. exists, and
3. .

It is continuous on the open interval (a,b) if it is continuous at each point in the interval.

We can define continuity from the left at x=c or continuity from the right at x=c by replacing the limits above by their one-sided versions.

The intuition behind the concept of continuity of f at c is that, in the vicinity of x=c, one can draw the graph of f without lifting one's pencil from the page. If a function is not continuous at a point, then it is discontinuous there.

Discontinuities come in various forms:

• holes,
• vertical asymptotes, and
• jumps

are the typical cases. Holes are somewhat different from the other two cases, because by simply filling the hole we can turn the point of the discontinuity into a point of continuity.

Examples: #1, 3, 4, 6, p. 164

Continuity on a closed interval: A function is continuous on a closed interval [a,b] if

1. it is continuous on the open interval (a,b),
2. it is continuous from the right at x=a, and
3. it is continuous from the left at x=b.

All types of functions we've considered to this point are continuous on their domains:

• polynomials
• power functions
• rational functions
• exponentials
• logarithms

So limits are easy to calculate for these functions: if you need the limit at c, simply calculate f(c) (if possible).

Examples: #7, 11, p. 164

Examples: #18, 24, 27, p. 164

Intermediate Value Theorem: if f is continuous on [a,b], f takes on every value between f(a) and f(b).

The idea behind the IVT is that, if you've got to connect from f(a) and f(b), and you can't lift the pencil, then you've got to go through the whole spectrum of values from f(a) to f(b).

Examples: #29, p. 165; 34, p. 165