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.
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 one-sided limit from the positive direction and the one-side limit from the negative direction are defined, but not equal.
Essential discontinuity, One or both of the one-sided limits does not exist or is infinite.
Estudy the continuity on R of the function: