Real Analysis

Date: 9/13/96 at 6:58:13
From: Anonymous
Subject: Real Analysis

Let I be a subset of R where I is an interval and let f map I to R be 
increasing on I. If c is not an end point of I, show that the jump of 
f at c is given by inf{f(y)-f(x): x<c<y, x, y is an element of I}.

Date: 9/13/96 at 13:18:57
From: Doctor Tom
Subject: Re: Real Analysis

I can't help you on this because I don't know what you mean
by "the jump of f at c".  It's not a standard term (as far as
I know), and if I were trying to make a definition, I would
probably define it using exactly the words you did in your
statement of what's to be proved.  That is, I would have said:

Definition:  The "jump" of f at c is:
             inf{f(y)-f(x): x<c<y, x, y is an element of I.

So if you can tell me the definition of "jump", I might be
able to help.

Or maybe you're asking why this makes sense as a definition?

The reason is that an increasing function can have any
countable number of discontinuities in the neighborhood of
a point, so in any interval around c, the discontinuitites
can be dense, so you have to use the inf definition.  It
simply says to try all combinations of values of f above 
and below c and find out how big the jump is between them.

Then you find an infimum, or lower bound for those numbers.
Obviously, you can't use any of the standard continuity
arguments because there may be an infinite number of discontinuities
in any interval around c.

Anyway, if this doesn't help, write again, but tell me the
"official" definition of "jump".

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 9/13/96 at 15:18:12
From: Barry Moore
Subject: Re: Real Analysis

The definition of "Jump" is :

The Jump of f at c is defined to be (the right hand limit of f) 
minus (the left hand limit of F at), i.e. if f is an increasing 
function on[a,b] and c in an element of the interval (a,b).

I hope this makes sense.

Barry

Date: 9/13/96 at 17:1:37
From: Doctor Tom
Subject: Re: Real Analysis

Okay. So one way to do this would be to show that there is a limit
on both the right and left.

Let me just look at the case where we approach c from below; the
case from above will be basically the same, perhaps with some
"greater-thans" replaced by "less-thans", and so on.

Since f is increasing, if x < c, f(x) is always less than f(c), so
all the f(x) values are bounded from above.  If I look at smaller
and smaller intervals below c, say between c and c-e, where e gets
smaller and smaller, the range of f(x) on those intervals gets
smaller and smaller.  I claim that this range must condense to a
point in the limit, because if the limit were a range, it has to
contain at least 2 values that f takes on, and simply choose e
small enough to chop out the lower one for a contradiction.

So a limit exists from below (and a similar agrument shows that
one exists from above).  The jump is the difference of those
values.

I claim that the inf of your expression is the difference of those
values.  If your inf is bigger, I can find an "a" below c and a "b"
above c so that f(b)-f(a) is arbitrarily close to your inf.  Pick
any smaller interval than from a to b, and any values in that interval
will give f(b)-f(a) values smaller than what you claimed to be the
inf (and closer to the difference of the limits), so the inf cannot
be different from the difference of the limits (which is the jump).

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/