With \( f(a) = 1 \) and \( f(b) = 3 \) there is no sign change, so the method cannot ensure a root. For example, \( f(x) = x^2 + 1 \) on \( [0, \sqrt{2}] \) has \( f(0) = 1 \) and \( f(\sqrt{2}) = 3 \), and no real root. Continuity alone is not enough without the sign change.
The bisection method is an iterative algorithm that approximates a root of a continuous function \( f(x) \) on an interval \( [a, b] \) with a sign change, halving the interval at each step.