
Continuity
A continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous.
Mathematically, function f is said to be continuous at point x = a if:
1. f(a) is defined, so that f(a) is in the domain of f.
2. exists for x in the domain of f.
3.
Why is this function not continuous at x=2?
This function is not continuous at x=2 because f(2) is not defined.
Discontinuities:
If a function is not continuous at a point in its domain, one says that it has a discontinuity there. There are different types of discontinuities:
 Removable discontinuity.If f(a) and are defined, but not equal.
 Jump discontinuity or step discontinuity, if the onesided limit from the positive direction and the oneside limit from the negative direction are defined, but not equal.
 Essential discontinuity, One or both of the onesided limits does not exist or is infinite.
Estudy the continuity on R of the function:
