Intermediate Value Property of Continuous Functions

Today's blog is another presentation of an elementary proof that I will use in my proof of the fundamental theorem of calculus.

The content in today's blog is taken from Edwards & Penney's Calculus and Analytic Geometry.

Theorem: Intermediate Value Property of Continuous Functions

If a function f is continuous on [a,b] and f(a) is less than K which is less than f(b), then K = f(c) for some number c in (a,b).

Proof:

(1) Let us start by defining three sequences {a_i}, {b_i}, and {I_i}

a₁ = a
b₁ = b

m_i = midpoint of [a_i,b_i]

we can assume that f(m_i) ≠ K since if it did, then c = m_i and we are done with this proof.

if K is less than f(m_i), then:

a_i+1 = a_i
b_i+1 = m_i

Otherwise K is greater than f(m_i) and:

a_i+1 = m_i
b_i+1 = b_i

In this way, we can also define a sequence of nested intervals such that:
I_i = [a_i,b_i] and for all i, K lies between f(a_i) and f(b_i)

(2) Using the Nested Intervals Property of Real Numbers (see Lemma 1, here), we know that there exists a point c in I₁ such that {c} = I₁ ∩ I₂ ∩ I₃ ∩ ...

(3) We can see that both {a_i} and {b_i} have this point c as their limit. [See Definition 1, here for a definition of a limit]

(4) Since the function f(x) is continuous on (a,b), we know (see Definition 1, here for definition of continuous on a point) that:

for any positive real ε, there exists a positive real δ such that:

if abs(x-c) is less than δ, then abs(f(x) - f(c)) is less than ε

(5) By the definition of limits of a function (see Definition 1, here), it follows that the limit of {f(b_n)} and the limit of {f(a_n)} is f(c).

(6) Since for all n, f(b_n) is greater than K, it follows that f(c) ≥ K.

(7) Likewise for all n, f(a_n) is less than K so that f(c) ≤ K.

(8) But the only way that step #6 and step #7 can both be true is if f(c)=K.

QED

References

• Edwards & Penney, Calculus and Analytic Geometry