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Let $f$ be a continuous function whose domain contains an open interval $(a, b)$. What form can $f(a, b)$ have?

Assume that $(a, b)$ is bounded. Does anyone know examples for the different forms this might take?

  • I think we could easily map this to just a single constant. E.g., $f(x)$ = $0$. Then $f(0,1)$ = {$0$}.

  • Or, we can easily map it to another open interval. E.g., $f(x)$ = $x$. Then $f(0,1)$ = $(0, 1)$.

Are there other possibilities?

  • 0
    Consider $\sin(0,10)$...2012-02-16
  • 0
    Think about where $\tan x$ sends the interval $(-\pi/2, \pi/2)$.2012-02-16

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