# Intermediate value theorem

If f is continuous on [a, b] and takes values with opposite signs at the ends, then it equals zero somewhere between them.

Maths (university) · Part 02: Limits and continuity · Card 8 of 9 · Last updated 2026-10-19

## Analogy

Hiking from a valley to a hilltop without teleporting: you must cross every height in between.

## Example

**Root of x^3 + x - 1**

f(0) = -1 and f(1) = 1.
Sign changes, f is continuous.
So a root lies in (0, 1).
f(0.5) = -0.375, so the root is in (0.5, 1).

## In practice

It proves that equations have solutions, and lets you narrow a root down by halving the interval.

## Practice

### Flashcards

- **What does the intermediate value theorem guarantee?** A continuous function with a sign change on [a, b] has a root in between.

## Questions people ask

### Why does the intermediate value theorem need continuity?

Without continuity the graph can jump over a value. The function 1/x is -1 at x = -1 and 1 at x = 1, yet it never equals zero, because it breaks at 0. Continuity guarantees there are no jumps, so every value in between is reached.

### Does the theorem tell me where the root is?

No, it only says a root exists inside the interval. To locate it, test the midpoint and keep the half where the sign changes. Repeating this is called the bisection method. Each round cuts the interval in half, so ten rounds shrink it by a factor of about a thousand.

## Related concepts

- [Continuity](https://studyvisually.com/learn/science/maths-university/maths-university-part-2/maths-university-part-2-concept-6)
- [Types of discontinuity](https://studyvisually.com/learn/science/maths-university/maths-university-part-2/maths-university-part-2-concept-7)
- [Differentiability](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-10)

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