The Stepping Stone Theorem
says that a function, f(x), is continuous at a point,
We show that if the sequence converges to a limit of c, then
as the subscripts grow, the xn's must fall within
smaller and smaller intervals of c. Another way of saying
that is that they will eventually fall within d of c,
no matter how close to zero you choose d to be. There is
not much to show here since this is all part of how we define what
it means for a sequence to converge to a limit. We then go on to
show that applying the definition of continuity to this leads to
the convergence of
If the sequence,
We also have...
If f(x) is continuous at c, then for any
|f(x) - f(c)| £ ewhenever
Whatever e I choose,
you can find a d small
enough to make the above inequality true. So every x within
d of c gives you an f(x) that is within
e of f(c).
But whatever that d
is, you can also find an n big enough so that xk
is within d of c whenever
Here we must show that if every set of stepping stones that leads
to the river bank has moss that leads to the moss on the riverbank, then
the moss
function is continuous. Or in other words, if the moss function is
not continuous, then at least one set of stepping stones that
leads to the riverbank has moss that does not lead to the moss on
the riverbank. That is the approach this proof takes. We assume
that the function, f(x), is discontinous at
First we define a sequence of d's. Let
1 d n =Assume now that f(x) is discontinuous atn
|x - c| £ dand
|f(x) - f(c)| > eIn other words, for f(x) to be discontinous at c, no matter how close I require an x to be to c, you can always find such an x that will make f(x) farther than e from f(c).
Since that rule exists for every d, that means
you can find such an x that is contained in the interval,
There is no question that the dn's
converge to
zero. That means that as n gets big, xn
gets squeezed closer and closer to c. In fact the
xn's must converge to c.
But remember that each xn was chosen so that
f(xn) would be farther than
e from
f(c). The discontinuity at c was what allowed you
to choose it that way. And because f(xn) never
gets within
e of f(c), we know that the sequence,
I don't expect that your instructor will ask you to reproduce the "if" part on an exam, but he or she may very well ask you to reproduce the "only if" part -- that is showing that f(x) being continuous at c implies that every sequence, xn, that converges to c generates a sequence, f(xn), that converges to f(c). Be sure you know what the theorem means, because can be worded in a number of different ways. And learn the proof.
I included the proof of the "if" part only to satisfy the curiosity of those who are interested.
One more optional item for the curiosity of those who are interested. We defined the property of continuity using a delta-epsilon contract. We have now shown that this converging sequence property given in the stepping stone theorem is true if and only if the continuity property is true. When you have one property that is true if and only if another is true, the two properties are said to be equivalent. That means that we could have chosen to define continuity using the converging sequence property, gone on to prove the delta-epsilon property, and everything else about continuity would remain the same.
email me at hahn@netsrq.com