MathDB

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Part of 1975 IMO Shortlist

Problems(1)

Is it possible to find a function f such that G(f)=G_a (f) ?

Source:

9/21/2010
Let f(x)f(x) be a continuous function defined on the closed interval 0≤x≤10 \leq x \leq 1. Let G(f)G(f) denote the graph of f(x):G(f)={(x,y)∈R2∣0≤f(x): G(f) = \{(x, y) \in \mathbb R^2 | 0 \leqx≤1,y=f(x)} x \leq 1, y = f(x) \}. Let Ga(f)G_a(f) denote the graph of the translated function f(x−a)f(x - a) (translated over a distance aa), defined by Ga(f)={(x,y)∈R2∣a≤x≤a+1,y=f(x−a)}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<1a, \ 0 < a < 1, a continuous function f(x)f(x), defined on 0≤x≤10 \leq x \leq 1, such that f(0)=f(1)=0f(0) = f(1) = 0 and G(f)G(f) and Ga(f)G_a(f) are disjoint point sets ?
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