Converse Of Intermediate Value Theorem

Calculus Level 2

A real-valued function f f is said to have the intermediate value property if for every [ a , b ] [a,b] in the domain of f f , and for every

x ∈ [ min ⁡ ( f ( a ) , f ( b ) ) , max ⁡ ( f ( a ) , f ( b ) ) ] , x \in \Big[\min\big(f(a), f(b)\big), \max\big(f(a), f(b)\big)\Big],

there exists some c ∈ [ a , b ] c\in [a,b] such that f ( c ) = x f(c) = x .

The intermediate value theorem states that if f f is continuous, then f f has the intermediate value property. Is the converse of this theorem true? That is, if a function has the intermediate value property, must it be continuous on its domain?

Yes No

Sameer Kailasa by Sameer Kailasa , 25, USA

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1 solution

Patrick Corn
Nov 19, 2016

The answer is no. The function f ( x ) = { sin ⁡ ( 1 / x ) if x ≠ 0 0 if x = 0 f(x) = \begin{cases} \sin(1/x) &\text{if } x \ne 0 \\ 0 &\text{if } x = 0 \end{cases} has the desired property, and is discontinuous at 0. 0.

Functions satisfying the intermediate value property are called Darboux functions ; Darboux proved that the derivative of a differentiable function is always a Darboux function, even if that derivative is discontinous.

3 Helpful 0 Interesting 0 Brilliant 1 Confused

(I know you didn't write this question, but I have some concerns about it)

  1. Should the property hold for every possible pair of endpoints a , b a, b , or just the endpoints of the domain? I'm thinking the former, since otherwise an "obvious discontinuity" counter example of f ( x ) : [ 0 , 2 ] → [ 0 , 1 ) f(x) : [0,2] \rightarrow [0,1) , f ( x ) = { x } f(x) = \{ x \} the fractional part function would suffice.

  2. Since the domain could be unbounded (and that the function need not be achieved), the max/min might not be achieved. Should min/max be replaced with open interval of inf/sup?

Calvin Lin Staff - 4 years, 6 months ago

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Good points. I imagine the intention was that the property was supposed to hold for any interval in the domain. So: " f f is a function on an interval, and for any a , b a,b in the domain of f , f, and any c c between f ( a ) f(a) and f ( b ) , f(b), there is an x ∈ [ a , b ] x \in [a,b] such that f ( x ) = c . f(x) = c. Is f f necessarily continuous on its domain?" That fixes 1 and 2, I think?

Patrick Corn - 4 years, 6 months ago

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