Continuity

Definition of Continuity: Three conditions for continuity of f(x) at a point x = c:

1. the function must be defined at c, or f(c) must be defined.
2. must exist.
3. 

Properties of continuous functions:

If two functions f(x) and g(x) are continuous at c, then

Sum of continuous functions: The sum of two continuous functions is continuous at c:
is continuous at c.

Constant * function: The product of a constant, k, and a continuous function is continuous at c:
is continuous at c.

Product of continuous functions: The product of continuous functions is a continuous function at c:
f * g is continuous at c.

Quotient of continuous functions: The quotient of continuous functions is a continuous function at c provided that the denominator is not 0:
is continuous at c if .

Composition of continuous functions: The composition of one continuous function by another is usually continuous at a point c:

is continuous if f is continuous at g(c).

Continuity of polynomials: A polynomial is continuous at every real number:
is continuous at every real number.

Removable discontinuity: A discontinuity in a function at a point is removable if the function can be made continuous by redefining the function at that point. I other words, if the limit exists at the point, simply redefine the value of the function to be equal to its limit at the point.

Intermediate Value Theorem: If f is continuous on an interval with y coordinates y1 and y2 at either end, then there is a point in the interval with a y coordinate in between these values.

In the graph, the end points have y coordinates of +1 and +4. Since the function is continuous, we can be sure that there is an x coordinate where the value of x is, say, 3 (any value between 1 and 4). That point is approximately (1,3).