Use the Intermediate Value Theorem to prove that ∃x, y ∈ R,(x < y ∧ f(x) = f(y))

Question

Let f : R → R be a continuous function. Suppose that

∃a, b ∈ R,(a < b ∧ f(a) < f(b))

and

∃c, d ∈ R,(c < d ∧ f(c) > f(d))

Use the Intermediate Value Theorem to prove that

∃x, y ∈ R,(x < y ∧ f(x) = f(y))

Hint: Construct a continuous function H(t) with the property that H(0) = f(b)−f(a) and H(1) = f(d) − f(c). Then use the Intermediate Value Theorem on H(t) over the interval [0, 1].

Answer 1

$H(t)=f((1-t)b+td)-f((1-t)a+tc)$.