MathDB

Let

f(x)

be a continuous function defined on the closed interval

0 \leq x \leq 1

. Let

G(f)

denote the graph of

f(x): G(f) = \{(x, y) \in \mathbb R^2 | 0 \leq

x \leq 1, y = f(x) \}

. Let

G_a(f)

denote the graph of the translated function

f(x - a)

(translated over a distance

a

), defined by

G_a(f) = \{(x, y) \in \mathbb R^2 | a \leq x \leq a + 1, y = f(x - a) \}

. Is it possible to find for every

a, \ 0 < a < 1

, a continuous function

f(x)

, defined on

0 \leq x \leq 1

, such that

f(0) = f(1) = 0

and

G(f)

and

G_a(f)

are disjoint point sets ?

Problem Statement