An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

30

Continuity of a function.

Bk. I. ch. V.

every value of x which differs from a by an arbitrarily small yet
finite quantity.
The functions
y x sin ( — ) and y = (sin( -) 2
Y -x-a \x a x-a x - aa
behave quite indeterminately; for whereas in each case the first factor
becomes arbitrarily great as x approximates to a, the second oscillates
between - 1 and + 1 in the first case, between 0 and + 1 in the
other, so that from no point however close to a does
f(a h)= -h sin(+ ) or +h (sin(+~ ))
present a series with a determinate finite or infinite limiting value as h
becomes infinitely small; neither the properties required in ~ 6 nor
those in ~ 9 are here fulfilled. The oscillations of the sine function
follow each other more rapidly the nearer x comes to a; if we assume
x = a + -h and ask: by how much h must be diminished in order that
the number under the sine may change by 2 r, it follows, putting
1 1
~ + 2 - _,
that
1 +2nh '
therefore
2 7th 2
h - hl —
A +2 2h
Thus the difference, the interval in which the sine travels all through
its values, is less than 2 t7 2, so that the number of oscillations
of the sine in an arbitrarily small region near a is far beyond any
assignable limit; for such a point, at which its argument becomes
infinite, the sine (and likewise the cosine, tangent and cotangent) has
no determinate value; therefore also it cannot be called continuous at
this point, although it is continuous at every point however near to it.
If a continuous function do not become infinite for any value
of x in an interval from a to b, then there is at least one value
of x in the interval or coinciding with either limit a or b, for which
it assumes a greatest value G assignable algebraically (that is, taking
account of sign) and likewise one at which it assumes an algebraically least value g. This proposition is not self-evident. In case of
a discontinuous function also, provided it remains finite, an upper
(and a lower) limit can be assigned, beyond which the values of the
function do not pass; moreover such limit can be fixed so that the
values of the function come arbitrarily near it at some point, and yet
the value itself never be actually reached. If we consider, for in