Expert: Paul Klarreich - 3/20/2006

Question
Hi, i emailed you a week or so ago with a problem and you got back to me with a solution but said you weren't entirlel satisfied with it and that one of your students may be able to get a better answer and i was just wondering whether they had done so yet as i'm still struggling with the problem. the questions was;

a function f: R -> R has the property that for any four real numbers a, b, c, d such that a - b > c - d, we have f(a) - f(b) > f(c) - f(d). prove that f is a linear function, ie f(x)  = mx + n for all x belonging to R, where m, n belong to R and m > 0

Thanks, Jake

Answer
Hi, Jake,

I have sent the problem on to one of my daughters.  (I no longer have students, just children.)  Her reply follows.  Anything I sent her is preceded by >'s or >>'s and anything I have added subsequently has ***s.

---------- the correspondence, slightly edited --------

That looks basically ok to me,

*** and believe me, if SHE says its ok, its ok.

although I have a few comments which I've interspersed in the text within double brackets. By the way, if possible
you should send your answer to the student in some other font than the one you used (eg courier), as I found it nearly impossible to distinguish between f and f' in this font.

*** Sorry -- I meant to add this:
WARNING: THE MATERIAL BELOW MAY CONTAIN FRACTIONS AND OTHER MATERIAL INAPPROPRIATE FOR CERTAIN COMPUTING SYSTEMS.  BE SURE TO VIEW IT IN A FIXED-SIZE FONT, SUCH AS COURIER.
***

Erica wrote:
>
>
> PKlarreich wrote:
>> Hi, Girls,
>>  
>> I promised to send this question on to you earlier, but never got
>> around to it.
>>  
>> --------- Student wrote -------------
>>  
>> Subject: Analysis
>> Question: a function f: R -> R has the property that for any four
>> real numbers a, b, c, d such that a - b > c - d, we have f(a) - f(b)
>> > f(c) - f(d). prove that f is a linear function, ie f(x) = mx + n
>> for all x belonging to R, where m, n belong to R and m > 0
>> ---------- I sent this answer  ------------------
>> If we assume that f'(x) exists for all x, then I can take a shot at
>> it like this:
>>
>> Assume f'(x) is not a constant. Then there are x1 and x2 where f'(x2)
>> > f'(x1),
>>
>> Write f'(x1) = m1 and f'(x2) = m2, and m2 > m1
>>
>> Write m2 = m1 + 3e
>>
>> Then (the Reverse Mean Value Theorem?) there exists some interval I2
>> about x2 where f'(x) > m1 + 2e.

[[Isn't this just continuity of f'? No need for the Mean Value Thm here.
Unless maybe you meant the Intermediate value thm? But I don't think you even need that. Just plain old continuity of f'. Of course, we haven't been given that f' is continuous (or even exists) but I bet the student was meant to assume that. For the purposes of this proof that's certainly an assumption.]]
>>
>> And there exists some interval I1 about x1 where f'(x) < m1 + e.
>>
>> Choose an interval [c,d] about x2 where:
>> c,d are both within I2
>> f(d) - f(c) = K

[[It seems to me that you need to be more careful about how K is chosen. It looks above as if you've simply chosen randomly a pair c and d within I2 and then defined K to be f(d)-f(c), but then what guarantee do you have that there will be a and b within I1 for which f(b)-f(a)=K? I think
you need to tighten that up, choose K sufficiently small that there exist a, b, c, d within the appropriate intervals such that... etc etc]]
>>
>>      f(d) - f(c)
>> Then ------------ > m1 + 2e
>>         d - c

[[Your fraction got messed up in transit, as you can probably see above -- it took me a while to figure out what you were trying to say. I think you should generally assume that things won't line up on the recipient's end the way the look on your end. Also, I think you should cite the Mean Value Thm to justify the above inequality.]]
>>
>> d - c < K/(m1 + 2e)
>>
>> Now choose an interval [a,b] about x1 where
>> a,b are both within I1
>> f(b) - f(a) = K
>>
>>      f(b) - f(a)
>> Then ------------ < m1 + e
>>        b - a
>>
>> So b - a > K/(m1 + e) > K(m1 + 2e) > d - c
>>
>> So we have b - a > c - d, but WE DO NOT HAVE
>>
>> f(b) - f(a) > f(d) - f(c) [they are equal]
>> --------------------
>>  
>> What do you think?  Is this any good?  Of course the student's
>> question did not make any assumptions about f(x), such as continuity,
>> differentiability, etc.
>>