Continuous functions have an essential role in calculus . Nearly all of the rules for the calculus operations assume that the function to be operated on is continuous . A function is continuous on a given interval if its graph can be traced without lifting the pencil from the paper. ++++ Mathematically , a function '? is continuous at x = a if 0 ?(a) lim m) x)a lim ?(x) = m) I Polynomial functions are continuous on all real numbers . Rational functions are continuous on the domain, where the denominator does NOT equal 0. Example: Determine whether the function ?(x) = 3x + 1 is continuous at x = 2