# Continuity on a Closed Interval

A function f(x) is continuous at the closed interval [a, b] if:

f(x) is continuous at x for all values of x belonging to the open interval (a, b).

f(x) is right-continuous:

f(x) is left-continuous:

**If f is continuous at a closed interval [a, b], then f is bounded on that interval.**

Study the continuity of at the interval [0, 4].

f(x) is left-continuous at x = 0 , since f(x) = x^{2} and polynomial functions are continuous at all of .

f(x) is right-continuous at x = 4 , since f(x) = 4 and polynomial functions are continuous at all of .

Study the continuity at x = 2, as this is the interval of the piecewise function. For f(x) to be continuous, it needs to be continuous at this point.

f(2)= 4

Thus, f (x) is continuous in the interval [0, 4].