Continuity of functions in an interval

A function f(x) is said to be continuous on an open interval (a,b) iff it is continuous at every point on the interval (a,b).

A function f(x) is said to be continuous on a closed interval [a,b] iff

f is continuous at every point on the interval (a,b), i.e., f is continuous on the open interval (a,b) and

lim (x→a⁺) f(x) = f(a) and lim (x→bˉ) = f(b).

It has to be continuous on (a,b), it has to be continuous at a from right and at b from left.