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
is defined, so that f(a)
is in the domain
exists for x in the domain
Why is this function not continuous at x=-2?
This function is not continuous at x=-2 because f(-2) is not defined.
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: