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A monotone function never has saddle points, same is true for an invertible function. Can we conclude that monotonic function is also invertible and vice-versa?

Peter_Pan
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kaka
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  • It might be a good idea to add a couple of details to your question: From context (including MPW's answer) it seem you're asking about real-valued functions of one real variable. Are you also assuming the domain is an interval? If not, then no, invertibility does not imply monotonicity. – Andrew D. Hwang Apr 03 '15 at 11:57
  • @user86418 domain being an interval is not enough unless you also assume continuity – Mirko Apr 03 '15 at 11:59
  • yes, domain is interval and function is continuous. Would it then be vice-versa? – kaka Apr 03 '15 at 11:59
  • use the intermediate value theorem – Mirko Apr 03 '15 at 12:07

1 Answers1

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Your statement

A monotone function never has saddle points

is false. The function $f:\mathbb R\to \mathbb R$ with $f(x)=x^3$ is (strictly) monotone, has a saddle point at $x=0$, and is invertible with inverse $f^{-1}(y)= y^{1/3}$.

Still, a strictly monotone function $g:\mathbb R\to \mathbb R$ is invertible with its inverse defined everywhere on $g(\mathbb R)$.

Strict monotonicity is required for invertibility. Nondecreasing step functions show why.

MPW
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    $f$ is invertible when restricted to any subset of its domain. The point is that is it is false to say that monotone functions cannot have saddle points. They can. – MPW Apr 03 '15 at 11:53
  • so monotonicity implies invertibililty and vice-versa? – kaka Apr 03 '15 at 11:55
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    the vice-versa is not true and not addresses in the above answer – Mirko Apr 03 '15 at 11:56
  • @kaka Invertible + continuous implies monotone – 5xum Apr 03 '15 at 12:10
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    @5xum you also need to assume that the domain is connected – Mirko Apr 03 '15 at 12:12
  • @Mirko Yes, I was implicitly speaking about functions $f:\mathbb R\to\mathbb R$. – 5xum Apr 03 '15 at 12:15
  • Note that a monotone function $\mathbb R\to \mathbb R$ needn't be increasing or decreasing. Continuity is required to force that. Monotonicity of $f:X\to Y$ just means that $f^{-1}(y)$ is connected for each point $y\in Y$. – MPW Apr 03 '15 at 12:24
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    Note also that monotone and monotonically increasing (or monotonically decreasing) don't mean the same thing. That's a technical distinction, and I realize that sometimes people say "monotone" when they mean "monotonically increasing". Perhaps that's what OP really meant. – MPW Apr 03 '15 at 12:35