There must be a zero between a two intervals of a continuous function y = f(x) as stated by intermediate value theorem. Intermediate value theorem states that for any continuous function y = f(x) with intervals ranging from (a, b), it takes any value of the between f (a) and f (b). Apart from taking any value it also has at least one value of the same function between the two intervals. To understand this, the theorem describes the function like one going through opposite sides of a line with one side being a negative and the other side being a positive side. By passing through these points to the other there must be a zero in between to separate the two sides (Shamseddine & Berz, 2007).
In the situation where signs f (a) and f (b) are same, the function will not have a zero in between the two lines. Reason being that the function does move from one side to another of the signs. A continuous function may pass through values of the same sign from one point to another therefore not guaranteeing that there will be a zero in between the point’s f (a) and f (b) (Shamseddine & Berz, 2007).
Shamseddine, K., & Berz, M. (2007). Intermediate value theorem for analytic functions on a Levi-Civita field. Bulletin of the Belgian Mathematical Society-Simon Stevin, 14(5), 1001-1015.