Definitions [4]
Definition: Continuous Function
A function f(x) is said to be continuous at a point x = a, if the following three conditions are satisfied
- f is defined at every point on an open interval containing a.
- \[\lim_{x\to a}f\left(x\right)\] exists.
- \[\lim_{x\to a}f\left(x\right)=f\left(a\right)\].
Definition: Discontinuous Function
A function f(x) is said to be discontinuous at x = a if it is not continuous at x = a, i.e.
- \[\lim_{x\to a}f\left(a\right)\] does not exist.
- The left-hand limit and the right-hand limit are not equal.
- \[\lim_{x\to a}f\left(x\right)\neq f\left(a\right)\].
Definition: Removable Discontinuity
If \[\lim_{x\to a^{-}}f\left(x\right)=\lim_{x\to a^{+}}f\left(x\right)\neq f\left(a\right),\] then f(x) is said to be removable discontinuous.
Definition: Non Removable Discontinuity
If \[\lim_{x\to a^{+}}f\left(x\right)\neq\lim_{x\to a^{-}}f\left(x\right),\] then f(x) is said to be non-removable discontinuous.
