Intermediate value theorem
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Contents
Statement
Let [ilmath](X,\mathcal{ J })[/ilmath] be a connected topological space and suppose [ilmath]f:X\rightarrow[/ilmath][ilmath]\mathbb{R} [/ilmath] is a continuous function, then^{[1]}:
- [ilmath]\forall x_1,x_2\in X\forall r\in\mathbb{R}\Big[\big(\text{Min}(f(x_1),f(x_2))\le r\le\text{Max}(f(x_1),f(x_2))\big)\implies\exists x\in X[f(x)\eq r]\Big][/ilmath]^{[Note 1]}
- In words: for any [ilmath]x_1,\ x_2\in X[/ilmath] and any [ilmath]r\in\mathbb{R} [/ilmath], if [ilmath]r[/ilmath] is between [ilmath]f(x_1)[/ilmath] and [ilmath]f(x_2)[/ilmath] then there exists an [ilmath]x\in X[/ilmath] such that [ilmath]f(x)[/ilmath] equals [ilmath]r[/ilmath].
Proof
- Let [ilmath]x_1,\ x_2\in X[/ilmath] be given
- Let [ilmath]r\in\mathbb{R} [/ilmath] be given
- Define [ilmath]m:\eq\text{Min}(f(x_1),f(x_2))[/ilmath] and let [ilmath]M:\eq\text{Max}(f(x_1),f(x_2))[/ilmath]
- Suppose that [ilmath](m\le r)\wedge(r\le M)[/ilmath] is false, that it doesn't hold. Then by the nature of logical implication the implication evaluates to true regardless of the truth or falsity of the RHS
- Suppose that [ilmath](m\le r)\wedge(r\le M)[/ilmath] holds, by the nature of logical implication we must show that in this case [ilmath]\exists x\in X[f(x)\eq r][/ilmath] for the implication to be true
- By the image of a connected topological space is connected we see that, as [ilmath]X[/ilmath] is connected, so is [ilmath]f(X)\subseteq\mathbb{R} [/ilmath]
- By a non-empty subset of the reals is connected if and only if it is a singleton or an interval we see that [ilmath]f(X)[/ilmath] is an interval if we consider [ilmath][a,a][/ilmath] as [ilmath]\{a\} [/ilmath] - which is sensible.
- As [ilmath]m\in f(X)[/ilmath] and [ilmath]M\in f(X)[/ilmath] we see that [ilmath][m,M]\subseteq f(X)[/ilmath] - note this may be a singleton or it may be an interval
- We can perform the rest of the analysis using case analysis
- Notice first:
- [ilmath][m,M]\subseteq f(X)[/ilmath] may be a singleton, but it may not be
- we are considering the case where [ilmath]r\in[m,M][/ilmath] this is okay because we know [ilmath]m\le r\le M[/ilmath] - which is the very definition of [ilmath]r\in[m,M][/ilmath] - see closed interval for more.
- The case analysis follows, we have 2 main cases: [ilmath][m,M][/ilmath] is a singleton and it is not, then some cases on [ilmath]r[/ilmath]^{[Note 2]}
- Suppose [ilmath][m,M][/ilmath] is a singleton
- then as [ilmath]r\in[m,M][/ilmath] we must have [ilmath]r\eq m[/ilmath] and [ilmath]r\eq M[/ilmath] (as [ilmath]m\eq M[/ilmath] for it to be a singleton)
- We see next that [ilmath]r\eq f(x_1)[/ilmath] and [ilmath]r\eq f(x_2)[/ilmath]
- Choose [ilmath]x:\eq x_1[/ilmath] ([ilmath]x_2[/ilmath] would have worked)
- We see that [ilmath]f(x)\eq r[/ilmath] as required for our choice of [ilmath]x[/ilmath]
- Choose [ilmath]x:\eq x_1[/ilmath] ([ilmath]x_2[/ilmath] would have worked)
- We see next that [ilmath]r\eq f(x_1)[/ilmath] and [ilmath]r\eq f(x_2)[/ilmath]
- then as [ilmath]r\in[m,M][/ilmath] we must have [ilmath]r\eq m[/ilmath] and [ilmath]r\eq M[/ilmath] (as [ilmath]m\eq M[/ilmath] for it to be a singleton)
- Suppose [ilmath][m,M][/ilmath] is not a singleton. We create 3 cases: [ilmath]r\eq m[/ilmath], [ilmath]r\eq M[/ilmath] and [ilmath]r\in(m,M)[/ilmath]
- [ilmath]r\eq m[/ilmath]
- This means [ilmath]r\eq f(x_i)[/ilmath] for some [ilmath]i\in\{1,2\} [/ilmath], as [ilmath]m:\eq\text{Min}(f(x_1),f(x_2))[/ilmath] remember.
- Choose [ilmath]x:\eq x_i[/ilmath]
- We see that [ilmath]f(x)\eq f(x_i)\eq r[/ilmath] so [ilmath]f(x)\eq r[/ilmath] as required
- Choose [ilmath]x:\eq x_i[/ilmath]
- This means [ilmath]r\eq f(x_i)[/ilmath] for some [ilmath]i\in\{1,2\} [/ilmath], as [ilmath]m:\eq\text{Min}(f(x_1),f(x_2))[/ilmath] remember.
- [ilmath]r\eq M[/ilmath]
- This means [ilmath]r\eq f(x_i)[/ilmath] for some [ilmath]i\in\{1,2\} [/ilmath], as [ilmath]M:\eq\text{Max}(f(x_1),f(x_2))[/ilmath] remember.
- Choose [ilmath]x:\eq x_i[/ilmath]
- We see that [ilmath]f(x)\eq f(x_i)\eq r[/ilmath] so [ilmath]f(x)\eq r[/ilmath] as required
- Choose [ilmath]x:\eq x_i[/ilmath]
- This means [ilmath]r\eq f(x_i)[/ilmath] for some [ilmath]i\in\{1,2\} [/ilmath], as [ilmath]M:\eq\text{Max}(f(x_1),f(x_2))[/ilmath] remember.
- [ilmath]r\in(m,M)[/ilmath]
- As [ilmath](m,M)\subseteq[m,M]\subseteq f(X)[/ilmath] we see [ilmath](m,M)\subseteq f(X)[/ilmath]^{[Note 3]}
- so [ilmath]r\in f(X)[/ilmath]
- By definition of image, [ilmath]r\in f(X)\iff \exists p\in X[f(p)\eq r][/ilmath]
- Choose [ilmath]x:\eq p[/ilmath] that we know to exist from [ilmath]r\in f(X)\iff \exists p\in X[f(p)\eq r][/ilmath]
- Now [ilmath]f(x)\eq r[/ilmath] as required.
- Choose [ilmath]x:\eq p[/ilmath] that we know to exist from [ilmath]r\in f(X)\iff \exists p\in X[f(p)\eq r][/ilmath]
- By definition of image, [ilmath]r\in f(X)\iff \exists p\in X[f(p)\eq r][/ilmath]
- so [ilmath]r\in f(X)[/ilmath]
- As [ilmath](m,M)\subseteq[m,M]\subseteq f(X)[/ilmath] we see [ilmath](m,M)\subseteq f(X)[/ilmath]^{[Note 3]}
- [ilmath]r\eq m[/ilmath]
- Suppose [ilmath][m,M][/ilmath] is a singleton
- We see that in all cases (and their sub-cases) that: [ilmath]\exists x\in X[f(x)\eq r][/ilmath], as required.
- Note: We could have tidied up the cases a bit, however we chose not to, as per^{[Note 2]}.
- Notice first:
- By a non-empty subset of the reals is connected if and only if it is a singleton or an interval we see that [ilmath]f(X)[/ilmath] is an interval if we consider [ilmath][a,a][/ilmath] as [ilmath]\{a\} [/ilmath] - which is sensible.
- We have seen that if [ilmath](m\le r)\wedge(r\le M)[/ilmath] that [ilmath]\exists x\in X[f(x)\eq r][/ilmath] indeed holds
- By the image of a connected topological space is connected we see that, as [ilmath]X[/ilmath] is connected, so is [ilmath]f(X)\subseteq\mathbb{R} [/ilmath]
- The implication holds
- Define [ilmath]m:\eq\text{Min}(f(x_1),f(x_2))[/ilmath] and let [ilmath]M:\eq\text{Max}(f(x_1),f(x_2))[/ilmath]
- Since [ilmath]r\in\mathbb{R} [/ilmath] was arbitrary the implication holds for all such [ilmath]r[/ilmath]
- Let [ilmath]r\in\mathbb{R} [/ilmath] be given
- Since [ilmath]x_1,x_2\in X[/ilmath] were arbitrary the implication holds for all such [ilmath]x_1,x_2[/ilmath]
This completes the proof.
Notes
- ↑ Recall that [ilmath]a\le b\le c[/ilmath] really means [ilmath](a\le b)\wedge(b\le c)[/ilmath] - so [ilmath]a\le b[/ilmath] and [ilmath]b\le c[/ilmath].
- ↑ ^{2.0} ^{2.1} We could get rid of one of our cases by noting that [ilmath]m\in[m,M][/ilmath] regardless of whether or not it is a singleton, we didn't do this to keep the "purity" of the case analysis
- ↑ Really in this case we have [ilmath](m,M)\subset[m,M][/ilmath] but [ilmath]\subseteq[/ilmath] still works, remember to claim it's a proper subset means to claim they're not equal, we do not bother with this here, even though it is true
References
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