Geometric continuity

From Freepedia

A function f(x) can be certain to have continuity in the interval [a,b] if an arbitrary point p in the interval has the following properties:

1. lim as x approaches p from the right side of f(x) is equal to z
2. lim as x approaches p from the left side of f(x) is also equal to z

Hence the point p is continuous for f(x) and if arbitrary point p is in the interval, then the interval [a,b] is continuous.

This definition may not be too practical in finding the continuity of a function f(x) in any interval since testing each and every point on the graph would be impossible. Therefore there are a set of rules in determining whether or not the function f(x) is continuous in any interval, and that is if there are no discontinuities in the function. Discontinuities are defined are either:

1. Removable discontinuities: f(x)=(x^2-x-2)/(x-2) { f(x) is undefined at x=2
2. Infinite discontinuities: f(x)=1/(x^2) { f(x) approaches infinity at x=0
3. Jump discontinuities: f(x)=x { f(x) is 0 at 0<=x<1 and 1 at 1<=x<2 and so on, the function jumps

Note: f(x)=x is called the greatest integer function.

Hence, absences of these discontinuities would imply that the function is continuous.