Intermediate Value Theorem We're looking at the intermediate value theorem. We will apply the idea of continuity to thefirst of these theorems in calculus. What is Intermediate Value Theorem in Math? Intermediate value theorem (IVT) is a basic theorem of calculus that says that if f(x) iscontinuous on the closed interval [a, b] and differentiable on the open interval (a, b), thenthere exists at least one c such that f(c) = 0. In other words, if you have a continuous function on a closed and bounded interval, thenthere will be at least one point where its derivative is zero. Try to connect these two dots with a continuous function that doesn't cross through thex-axis. The problem is that this is not a continuous function. Once again, we've created avertical asymptote. This isn't continuous, either. The function fails the vertical line test, since it does not stay on the x-axis as it goes throughthe origin; moreover, if you change the rules for how it stays on the x-axis, then you havemoved into three dimensions and cannot possibly call this an x-axis anymore. The intermediate value theorem is the primary reason for this phenomenon.
The theorem that describes this concept is called the Intermediate Value Theorem, whichstates that if a variable has an x-value between two other numbers, a and b, then thevariable must also have a y-value between these same two numbers. Any number within therange of a and b must fall within the range of f(a) and f(b).Furthermore, the reason this task is impossible is because you are asked to count fromnegative to positive values along the x-axis, which goes through 0. The intermediate value theorem states that if a continuous function on an interval [a, b] takesthe same value at two points in the interval, then it will take every value between these twovalues. The intermediate value theorem states that if f of x is continuous on a closed interval [a, b],then f(x) must take on all values between f(a) and f(b).