Analysis: Continuous open mappings.

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Here is a mystifying question from Rudin Chapter 4, #15

Call a mapping of X into Y "open" if f(V) is an open set in Y whenever V is an open set in X. Prove that every continuous open mapping of Reals into the Reals is monotonic.​

I'm having trouble proving this, in part, because I don't even think it's true. Wouldn't say... [tex]f(x)= x^{2}[/tex] map open sets to open sets? And [tex]f(x)= x^{2}[/tex] isn't monotonic on the Reals. Can someone tell me why [tex]f(x)= x^{2}[/tex] isn't a continuous open mapping of Reals into the Reals that is NOT monotonic.
 
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morphism
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I'm having trouble proving this, in part, because I don't even think it's true. Wouldn't say... [tex]f(x)= x^{2}[/tex] map open sets to open sets? And [tex]f(x)= x^{2}[/tex] isn't monotonic on the Reals. Can someone tell me why [tex]f(x)= x^{2}[/tex] isn't a continuous open mapping of Reals into the Reals that is NOT monotonic.
What is f((-1,1))?
 
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