An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

CONTINUOUS AND DISCONTINUOUS.

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11. The functions which commonly occur in mathematical The functions which
investigations are of such a nature, that they always suffer a commonly
occur in
gradual and not a sudden change, so to speak, when themathemavariable is gradually altered in value. Functions of this kind tics eous
are called Continuous Functions. By saying that a function
f(x) always receives a gradual change when x is gradually
varied, we mean this; that if x be changed into a', and therefore f (v) into f ('), then f(') -f( () may be made as small
as we please by taking x' - x' small enough, whatever be the
value of x. That this is true for all ordinary functions, such
as Ix, ax, log., sin x, &c.... &c. and all ordinary combinations
of these functions, such as (ax + log sin x)", ax sin x, &c...&c.
does not require to be proved, for it is quite evident. All
ordinary functions therefore are continuous functions.
12. Functions which sometimes suffer a sudden change Discontiwhen the variable is gradually altered in value, are called nuous
functions.
Discontinuous Functions. Thus, if we draw a broken curve
as in (fig. 3) the ordinate will be a discontinuous function of
the abscissa: for it is evident that the ordinate will suffer a
sudden change at the points P, Q, R, S, supposing the abscissa to be gradually varied.
13. We shall never have occasion to consider discontinuous functions in the following pages, and therefore we shall
always suppose that the functions we make use of are continuous.
Hence, if f(x) be any furction of x we make use of, and Assumption which
if x be changed into x', and therefore f(x) into f(i'); wewe shall
shall always assume that f(m) f(x) may be made as small kalas
as we please by taking x' - small enough whatever be the respecting
value of x.
14. A function f (x) is said to become Illusory when the An illusory
function
operations represented by f cease to give any definite result; defined.
which may happen in certain cases, as we shall shew.
X *' vx denotes the difference between x' and x, subtracting the greater of these
quantities from the lesser; x' r x is therefore the absolute difference between x' and x
without regard to sign.