Intermediate value theorem

Key takeaways

Intermediate value theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Intermediate value theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Intermediate value theorem from memory before moving on to harder problems.

Reference excerpt

In mathematical analysis, the intermediate value theorem states that if f {\displaystyle f} is a continuous function whose domain contains the interval [a, b] and s {\displaystyle s} is a number such that f ( a ) < s < f ( b ) {\displaystyle f(a)<s<f(b)} , then there exists some x {\displaystyle x} between a {\displaystyle a} and b {\displaystyle b} such that f ( x ) = s {\displaystyle f(x)=s} . That is, the image of a continuous function over an interval is itself an interval that contains f ( a ) , f ( b ) {\displaystyle f(a),f(b)} . For example, suppose that f ∈ C ( [ 1 , 2 ] ) , f ( 1 ) = 3 , f ( 2 ) = 5 {\displaystyle f\in C([1,2]),f(1)=3,f(2)=5} , then the graph of y = f ( x ) {\displaystyle y=f(x)} must pass through the horizontal line y = 4 {\displaystyle y=4} while x {\displaystyle x} moves from 1 {\displaystyle 1} to 2 {\displaystyle 2} . Over the interval, the set of function values has no gap, and the graph can be drawn without lifting a pencil from the paper. The corollary Bolzano's theorem states that if a continuous function has values of opposite sign inside an interval, then it has a root in that interval. The theorem depends on, and is equivalent to, the completeness of the real numbers, although Weierstrass Nullstellensatz is a version of the intermediate value theorem for polynomials over a real closed field. A similar result to the intermediate value theorem is the Borsuk–Ulam theorem, which underpins why rotating a wobbly table will always bring it to stability. Darboux's theorem states that all functions that result from the differentiation of some other function on some interval have the intermediate value property, even though they need not be continuous.

Motivation

This captures an intuitive property of continuous functions over the real numbers: given f {\displaystyle f} continuous on [ 1 , 2 ] {\displaystyle [1,2]} with the known values f ( 1 ) = 3 {\displaystyle f(1)=3} and f ( 2 ) = 5 {\displaystyle f(2)=5} , then the graph of y = f ( x ) {\displaystyle y=f(x)} must pass through the horizontal line y = 4 {\displaystyle y=4} while x {\displaystyle x} moves from 1 {\displaystyle 1} to 2 {\displaystyle 2} . It represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting a pencil from the paper.

Theorem The intermediate value theorem states the following: Consider the closed interval I = [ a , b ] {\displaystyle I=[a,b]} of real numbers R {\displaystyle \mathbb {R} } and a continuous function f : I → R {\displaystyle f\colon I\to \mathbb {R} } . Then

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Illustrations

Worked examples

Example 1 — a first encounter with Intermediate value theorem

Start with the simplest possible case. Write down what Intermediate value theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Intermediate value theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Intermediate value theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Intermediate value theorem

In research: Intermediate value theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Intermediate value theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Intermediate value theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Theorems in calculus, Theorems in real analysis, Theory of continuous functions, so understanding it makes those chapters shorter.
In everyday life: Look for Intermediate value theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Intermediate value theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Intermediate value theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Intermediate value theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Intermediate value theorem in simple terms?

In mathematical analysis, the intermediate value theorem states that if f {\displaystyle f} is a continuous function whose domain contains the interval [a, b] and s {\displaystyle s} is a number such that f ( a ) < s < f ( b ) {\displaystyle f(a)<s<f(b)} , then there exists some x {\displaystyle x}…

Why does Intermediate value theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Intermediate value theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Intermediate value theorem.