Open map and topology

Question

If $ f:\mathbb{R} \to \mathbb{R}$, $f(x)= x^3+3x^2+ax+3$, for what $a$ is $f$ an open map?

I was thinking the following:

It suffices to show $f$ maps basic open set (interval) to an open set in $\mathbb{R}$. Since $f$ is a continuous map because it is a polynomial, $f$ should map an open interval into a connected subset of $\mathbb{R}$ which is of the form of an interval. But what are the other conditions needed to make sure it is an open map?

If $f$ has a local max or min, it is not open. Can you see why? — Cheerful Parsnip, Apr 24 '13 at 01:56
@GrumpyParsnip because it maps into a half-open interval? So as long as I require that $f' \ne 0$ than is enough? — user74074, Apr 24 '13 at 01:58
@Grumpy Parsnip...Hi, cant see why. Can you give a short explanation? Thanks. — JKnecht, Feb 07 '16 at 06:33
@JKnecht: take a local max $f(x_0)$. Let $U$ be a small neighborhood of $x_0$ so that $f(x_0)$ is the max of $f$ on $U$. Then $f(U)$ is not open, since it has a maximum point. — Cheerful Parsnip, Feb 07 '16 at 15:07

score 2 · Answer 1 · answered Apr 24 '13 at 02:09

2

This is an application of a common question which states: "prove a continuous monotonic function is an open map" or "prove a strictly monotonic onto function has a continuous inverse".

Essentially, you want the derivative always positive or always negative, and you can find conditions on $a$ for that.

answered Apr 24 '13 at 02:09

toypajme

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Hi toypajme: +1) Nice. – Paul May 02 '13 at 07:49

Open map and topology

1 Answers1