Continuity Over an Interval

A function f(x) is said to be continuous in its domain if it is continuous at every point in its domain.

• A function f(x) is said to be continuous in an open interval (a, b) if it is continuous for every value of x in the interval (a, b).
• A function f(x) is said to be continuous in the closed interval [a, b] if
  (a) It is continuous for every value of x in the open interval (a, b).
  (b) f(x) is continuous at x = a from right
  i.e.\[\lim_{x\to a^{+}}f\left(x\right)=f\left(a\right)\]
  (c) f(x) is continuous at x = b from left,
  i.e. \[\lim_{x\to a^{-}}f\left(x\right)=f\left(b\right)\].