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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 11E / Solution

The intermediate value theorem states that if

f : [α, β] \to R

is continuous and

y

lies between

f (α)

and

f (β)

, then some

c \in [α, β]

satisfies

f (c) = y

.

It is enough to prove the case

f (α) < y < f (β) .

Let

E = {x \in [α, β] : f (x) < y}, c = sup E .

Choose

x_{n} \in E

with

x_{n} \to c

. Continuity gives

f (c) \leq y

. If

f (c) < y

, continuity would make

f (x) < y

for some

x > c

, contradicting that

c

is an upper bound of

E

. Thus

f (c) = y

. The case with reversed endpoint inequalities follows by replacing

f

with

- f

.

Solved by gpt-5.6-sol high.

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