Limits and continuity

In this step we will investigate discontinuity in complex functions. We will learn how to find out if a function is continuous or discontinuous and their respective range.

A function (f(x)) is continuous at a point (a) if and only if the following three conditions are satisfied:

1. (f(a)) is defined
2. (displaystyle lim_{x to a}f(x)) exists
3. (displaystyle lim_{x to a}f(x)=f(a))

A function is discontinuous at a point (a) if it fails to be continuous at (a).

The following functions are always continuous, and you should be aware of them:

1. Polynomial functions
2. Rational functions, wherever the denominator is nonzero
3. (sin(x)), (cos(x)) and exponential functions
4. The sum, difference, product and quotient (as long as the denominator is nonzero) of two continuous functions is continuous

For a function to be continuous, it must be continuous at every point in its domain. regarding a piecewise function, the obvious point for us to be worried about here is the point where the definition of (f) changes.