Intermediate Value Theorem (IVT)

Definition

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a,b], then it takes on every value between f(a) and f(b). In simpler terms, it guarantees that if you draw a continuous curve from one point to another without lifting your pencil off the paper, you will hit every y-value in between.

Related terms

Continuity:

A function is continuous if it has no breaks or jumps in its graph. It's like driving smoothly on a well-paved road without encountering any potholes.

Closed Interval:

A closed interval includes both endpoints. For example, [a,b] means it includes the values of a and b. It's like having a fence around your backyard that covers all the space between two posts.

Function:

A function is a rule that assigns each input value (x) to exactly one output value (y). It's like a vending machine that gives you only one type of candy for each button you press.