MathDB
epsilon-delta problem

Source: Romanian District Olympiad 2015, Grad XI, Problem 1

September 25, 2018
functionreal analysis

Problem Statement

Let f:[0,1]⟶[0,1] f:[0,1]\longrightarrow [0,1] a function with the property that, for all y∈[0,1] y\in [0,1] and ε>0, \varepsilon >0, there exists a x∈[0,1] x\in [0,1] such that ∣f(x)−y∣<ε. |f(x)-y|<\varepsilon .
a) Prove that if f∣[0,1] \left. f\right|_{[0,1]} is continuos, then f f is surjective. b) Give an example of a function with the given property, but which isn´t surjective.
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