A continuous function, f, has domain all real numbers. If f(-1) = 5 and f(1) = -5, explain why f must have at least one zero in the interval [-1, 1].

Answer 1

since f is continuous, there are no holes in the graph. Since f(-1) > 0 and f(1) < 0, f must assume all values between 5 and -5. That includes zero.