Convex function -> continuous?
Geoffrey T. Falk
closed subinterval [a,b] of (A,B) and for every point x in [a,b],
(x - a) (f(b) - f(a)) <= (b - a) (f(x) - f(a)).
I would like to see a proof or counterexample to the following:
If f is convex on (A,B), then f is continuous on (A,B).
This is *not* true if the domain is not open. (A counterexample is
the sign function on [0,1].)
Thanks
Geoffrey
--
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Geoffrey T. Falk, BSc MA; SCJ2P, SCSadm7, FreeBSD <gtf(@)cirp.org>
Peter L. Montgomery
>Let f: (A,B) \subset R --> R. Define f to be _convex_ when, for any
>closed subinterval [a,b] of (A,B) and for every point x in [a,b],
>
> (x - a) (f(b) - f(a)) <= (b - a) (f(x) - f(a)).
>
>I would like to see a proof or counterexample to the following:
>If f is convex on (A,B), then f is continuous on (A,B).
>
>This is *not* true if the domain is not open. (A counterexample is
>the sign function on [0,1].)
>
f(alpha*a + (1-alpha)*b) >= alpha*f(a) + (1 - alpha)*f(b)
when a, b in (A, B) and 0 <= alpha <= 1. Geometrically,
if (a, f(a)) and (b, f(b)) are two points on the graph of f,
then (x, f(x)) is on or above the line segment passing through
(a, f(a)) and (b, f(b)) whenever x is between a and b.
Look for a function with graph resembling (in fixed font)
--------------------
/
/
/
with a discontinuity where the diagonal line ends
and the horizontal line begins.
--
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Peter-Lawren...@cwi.nl Home: San Rafael, California
Microsoft Research and CWI
The World Wide Wade
gtfN...@NOSPAM.cirp.org (Geoffrey T. Falk) wrote:
> I would like to see a proof or counterexample to the following:
> If f is convex on (A,B), then f is continuous on (A,B).
Hint: Given b in (A,B), choose a and c in (A, B) with a < b < c. Now
consider the line through (a,f(a)) and (b,f(b)) and the line through
(b,f(b)) and (c,f(c)). To the right of b, f has to stay on or above the
first line and on or below the second line. To the left of b, an analogous
statement holds.
Stephen J. Herschkorn
>
>>Let f: (A,B) \subset R --> R. Define f to be _convex_ when, for any
>>closed subinterval [a,b] of (A,B) and for every point x in [a,b],
>>
>> (x - a) (f(b) - f(a)) <= (b - a) (f(x) - f(a)).
>>
>>I would like to see a proof or counterexample to the following:
>>If f is convex on (A,B), then f is continuous on (A,B).
>>
>>This is *not* true if the domain is not open. (A counterexample is
>>the sign function on [0,1].)
>>
>
>
> It is easy to transform this definition into the standard definition
>
> f(alpha*a + (1-alpha)*b) >= alpha*f(a) + (1 - alpha)*f(b)
>
>when a, b in (A, B) and 0 <= alpha <= 1. Geometrically,
>if (a, f(a)) and (b, f(b)) are two points on the graph of f,
>then (x, f(x)) is on or above the line segment passing through
>(a, f(a)) and (b, f(b)) whenever x is between a and b.
>
again...)
No matter: both concave and convex functions on open sets are
continuous. (Indeed, one of f or -f is convex iff the other is
concave.)
--
Stephen J. Herschkorn hers...@rutcor.rutgers.edu
Dave L. Renfro
[sci.math Dec 28 2002 12:50:29:000AM]
http://mathforum.org/discuss/sci.math/m/468956/468956
wrote (in part):
> I would like to see a proof or counterexample to the following:
> If f is convex on (A,B), then f is continuous on (A,B).
I'm sure that others will answer your specific question. Below
is part of an old post of mine that gives a summary of a few key
results about convex functions in case you or others are interested.
Dave L. Renfro
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Subject: [ap-calculus] Convex Functions
Author: Dave L. Renfro <dlre...@gateway.net>
Date: Thu, 14 Jun 2001 20:38:52 -0500
DEFINITION: Let f be defined on an interval I. We say that f
is convex on I if whenever x1, x2 belong to I, then
the line segment whose endpoints are (x1,f(x1)) and
(x2,f(x2)) lies on or above {(x,f(x)): x in [x1, x2]}.
If "on or above" is strengthened to "strictly above", we get a
geometric condition for concave up. Many of the results given
below continue to hold when f is concave up. Some of these will
be automatic (e.g. when the hypothesis includes "convex", since
concave up implies convex) and some of these will continue to hold
for other reasons. However, I don't really have the time or desire
right now to try and sort out which continue to hold when "convex"
is replaced with "concave up". <snip>
Here is another characterization of convex functions.
THEOREM : A function f is convex on an interval I if and only if
the following condition holds:
Whenever x1 < x2 < x3 belong to I, then
[f(x2) - f(x1)] / (x2 - x1) < or = [f(x3) - f(x2)] / (x3 - x2).
In other words, the average rate of change of f on
[x1, x2] does not exceed the average rate of change
of f on [x2, x3].
---------- SOME RESULTS ----------
Convex functions have many applications, both in pure mathematics
and in applied mathematics -- Many useful inequalities, including
the arithmetic and geometric mean inequality, can be obtained very
easily using convex functions. In recent years there has been a lot
of interest in convex functions defined on Banach spaces, especially
in their differentiability properties. Finally, convex functions play
a crucial role in linear programming and in optimization theory. For
more information, go to <http://www.google.com/> and search using
"differentiability +of convex functions" (quotes included),
"convex functions optimization" (quotes NOT included),
and "convex functions linear programming" (quotes NOT included).
1. If f is convex on an open interval I, then f is continuous at
each point in I.
2. If f is convex on an open interval I, then f is Lipschitz
continuous on each closed subinterval of I. [Lipschitz continuous
means the difference quotients are bounded.] This strengthens #1.
3. If f is convex on an open interval I, then there are at most
countably many points at which f is not differentiable.
Any countable set can be the set of non-differentiability points
for some convex function. Curiously, I couldn't find this in any
of the real analysis texts I looked at. However, this is a special
case of theorem 4.20 on p. 93 of [1].
4. Assume that f is convex on an open interval I. Then, at each point
of I, both the left derivative of f and the right derivative of
f exist. This strengthens #3. [It can be shown that this property
implies the property given in #3, but not conversely.]
5. If f is convex on an open interval I, then the left derivative
of f is a non-decreasing function, the right derivative of f
is a non-decreasing function, and at each point the left
derivative is less than or equal to the right derivative.
(See [7], p. 109.) This strengthens #4.
6. If f is convex on an open interval I, then the second derivative
of f exists at every point of I except for a set of measure zero.
[This follows from #3, #5, and the fact that monotone functions
are differentiable almost everywhere.]
7. If f is convex on an open interval I, and g is either the left
derivative of f or the right derivative of f (it doesn't matter
which one you let g be), then
f(b) - f(a) = integral of g on the interval [a,b]
for all a,b in I. [Monotone functions are Riemann-integrable,
so this is the usual calculus integral.]
8. Suppose f'' exists at each point of an open interval I. Then f is
convex on I if and only if f''(x) is nonnegative for each x in I.
9. Suppose f is continuous. Then f is convex on an open
interval I if and only if
limit as h --> 0 of [ f(x+h) + f(x-h) - 2*f(x) ] / (h^2)
is nonnegative for each x in I. [This strengthens #8, since
the existence of f'' at a point implies the limit above exists
at that point, and the converse fails.]
Riemann introduced and used this "second order symmetric
derivative" in an 1854 memoir on trigonometric series. It
was this memoir, incidentally, that Riemann introduced what
we now call the Riemann integral. LaTeX, .dvi, .ps, and .pdf
files of Riemann's 1854 memoir are available at
<http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Trig/>.
10. Suppose f' exists at each point of an open interval I. Then f is
convex on I if and only if f' is non-decreasing on I. This
strengthens #8 and neither implies nor is implied by #9.
11. If h is non-decreasing on an open interval I and 'a' belongs
to I, then the function f defined on I by
f(x) = integral of h on the interval [a,x]
is convex on I. [This refines a result that arises by putting
#5 and #8 together.]
---------- SOME REFERENCES ----------
[1] Yoav Benyamini and Joram Lindenstrauss, "Geometric Nonlinear
Functional Analysis", Volume 1, Colloquium Publications #48,
American Mathematical Society, 2000. [chapter 4: "Differentiation
of Convex Functions", pp. 83-98]
[2] Ralph P. Boas, "A Primer of Real Functions", 4'th edition
(revised and updated by Harold P. Boas), Carus Mathematical
Monographs 13, Mathematical Association of America, 1996.
[pages 175-186]
[3] Andrew M. Bruckner, "Differentiation of Real Functions",
CRM Monograph Series #5, American Mathematical Society, 1994.
[pages 131-134 (advanced)]
[4] Krishna M. Garg, "Theory of Differentiation", Canadian
Mathematical Society Series of Monographs and Advanced Texts
#24, John Wiley and Sons, 1998. [pages 195-198 (very advanced)]
[5] R. Kannan and Carole King Krueger, "Advanced Analysis on the
Real Line", Springer-Verlag, 1996. [pages 74-76]
[6] A.C.M. van Rooij and W.H. Schikhof, "A Second Course on Real
Functions", Cambridge University Press, 1982. [pages 14-18]
[7] H. L. Royden, "Real Analysis", 2'nd edition, MacMillan, 1968.
[pages 108-110]
[8] Brian S. Thomson, "Symmetric Properties of Real Functions",
Pure and Applied Mathematics #183, Marcel Dekker, 1994.
[pages 202-209 (advanced)]
[9] Richard L. Wheeden and Antoni Zygmund, "Measure and Integral",
Pure and Applied Mathematics #43, Marcel Dekker, 1977.
[pages 118-124]
Dave L. Renfro
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