INTERMEDIATE VALUE BUT DISCONTINUOUS EVERYWHERE
Alexander Abian
In what follows f is a real-valued function defined on the unit
closed interval [0, 1]. Also, by a subinterval we mean an interval
which has more than one point.
LEMMA 1. If the function f maps every subinterval of [0, 1]
onto [0, 1] then f is discontinuous at every point
x of [0, 1].
PROOF. By the hypothesis there is a sequence converging to x such
that at each point of that sequence, f assumes, say, 1. But
then there is also a sequence converging to x such that at
each point of that sequence f assumes, say, 0. Thus, f cannot
be continuous since otherwise f(x) = 1 =/= 0 = f(x).
In what follows by "between" we mean "equal or strictly in between"
DEFINITION. f is said to have the "intermediate value" property iff
for every real number r between f(a) and f(b) there
exists a real number c between a and b such that
f(c) = r.
LEMMA 2. If the function f maps every subinterval of [0, 1] onto
[0, 1] then f has the intermediate value property
PROOF. Trivial, since in every subinterval [a, b] of [0, 1] the
function f assumes every value between 0 and 1 and hence,
by the hypothesis, between f(a) and f(b).
THEOREM. There exists a function f which maps every subinterval
of [0, 1] onto [0, 1].
PROOF. There are some examples in the literature for f. Perhaps
the following is a simplest example.
We define f as follows.
(1) if the decimal expansion of x ends up with the pattern
.....0a0b0c0d0e0m0n...... with no digit a,b,c,...equal to 0
then we define f(x) = 0.abcdemn.......
For example if x = 0. 2617080239908090605080309090803030409020 .....
then f(x) = 0.89658399833492....
(2) otherwise f(x) = 0
To prove the Theorem it is enough to show that in every subinterval
[a, b] of [0, 1] and corresponding to any real number y in [0, 1],
for instance, say,
(3) y = 0.437689675998213.......
there exists a real number x in [a, b] such that
(4) f(x) = y
To this end, it is enough to observe that it is always possible to
find a point x in [a, b] which is arbitrarily close to a and such
that the decimal expansion of x from somewhere on ends up with
the pattern:
(5) x = ............ 0403070608090607050909080201030
But then by (1) we see that (5) and (3) imply (4), as desired.
A known Corollary follows:
COROLLARY. The intermediate value property does not imply continuity.
PROOF. The function f mentioned in the Theorem by Lemma 2 has the
intermediate value property, however by Lemma 1, it is discontinuous
everywhere in [0, 1].
In connection with the Corollary, we note,however, that it is well
know that every continuous function has the intermediate value property.
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ABIAN TIME-MASS EQUIVALENCE FORMULA T = A m^2 in Abian units.
ALTER EARTH'S ORBIT AND TILT TO STOP GLOBAL DISASTERS AND EPIDEMICS.
JOLT THE MOON TO JOLT THE EARTH INTO A SANER ORBIT.ALTER THE SOLAR SYSTEM.
REORBIT VENUS INTO A NEAR EARTH-LIKE ORBIT TO CREATE A BORN AGAIN EARTH(1990)
THERE WAS A BIG SUCK AND DILUTION OF PRIMEVAL MASS INTO THE VOID OF SPACE
Alexander Abian
NOTE the corrected (1*)
(1*) .....0a0b0c0d0e0m0n......