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.

Continuity of a function.

Bk. I. ch. V.

neighbourhood of this point, the following general theorems regarding
the property of continuity hold good:
a) The algebraic sum of two or more continuous functions is itself
a continuous function.
For if (p(x) and O(x) be continuous, it is always possible to discover
an interval h, such that
abs [p (x + lh) - p (x)] < ' 6, abs [( (x + ih) -- V (x)] <-.
From this it follows that
abs [1g(x +h) + (x +h)} {((x) Q(x)}] <.
b) The product of two or more continuous functions is itself a
continuous function.
For we have: abs [(p(x + h). 4 (x + h) -- p (x). -(x)]
= bs [p (x + h){ (x + ~) - (x)} + () (x){(x -+ h) - p (x)}].
If the interval h be so determined that
abs [~ (x + h) - ( (x)] < 6, abs [C (x + ) - p (x)] < E,
then we have:
abs [(p (x + ih) ( (x + h7) - gp (x) (x)] < E. abs [9p (x + h) + V (x)].
But since 9p and 4 have determinate upper limits in the neighbourhood
of the point under consideration, by assuming s suitably and determining h to correspond, this expression on the right can always be
made smaller than any prescribed small number.
c) The quotient of two continuous functions is itself a continuous
function, except at points at which the denominator vanishes.
For we have:
-as r^ (x +h) _ (x)7
abs
)p ( x- h) -(X)J
(L (x) p (x~ +)
A similar consideration to the last, shows that the numerator of this
expression can be made as small as we please, while the denominator
remains finite.
But if in the fraction qp(x) the denominator converges to zero for
) (X)
x = a, the numerator remaining finite, the value of the quotient becomes discontinuous, being either determinately infinite or indeterminate; though for every value of x however near this it is continuous.
If the numerator also converges to zero,'then by the value of
~(x) for x = a is to be understood the limiting value of the series
V (x)
obtained when x travels through a continuous series of numbers having
a for its limit. Whether the quotient then acquires a determinate