But for those who study calculus, it is an area of mathematics pioneered by Got-
tfried Leibniz. His idea was based purely on the concept of infinitesimals; this was in
opposition to the calculus of Isaac Newton, who based his calculus on the concept of
the limit. Although historically the emphasis was placed on the minute, modern infin-
itesimal calculus actually has little to do with infinitely small quantities.
What does continuousand discontinuousmean in calculus?
When talking about polynomial functions, we know that a polynomial function P(x)
satisfies the limit function:
in which arepresents all real numbers. This is called continuity.But if f(x) is a function on an interval around a,then f(x) is continuous at aiff (see
above for the definition of “iff”):
if it is not, then f(x) is called discontinuous at a.What is the concept of bound?
Similar to sequences (see elsewhere in this chapter), in calculus bounds are divided
into the upper (either greater than or equal to every other number in a set of num-
bers; or greater than or equal to all the partial sums of a sequence) or lower (less than
or equal to every other number). The symbol for infinity () is used to denote a set of
numbers without bound,or that increase or decrease “to infinity.” In calculus, the
bounds can be divided even more into greatest or least. For example, the greatest
lower bounds and least upper bounds are of special interest to calculus, as those num-
bers may or may not be found within a set.
xalim" fx()=fa()
xalim" Px()=Pa()
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MATHEMATICAL ANALYSIS
What does the strange symbol “iff” mean in calculus?
T
he symbol (or “word”) iffactually is shorthand for “if and only if.” It is not
only mathematics that depends on the iff, but also philosophy, logic, and
many technical fields. It is usually italicized; in addition, the phrase “P is neces-
sary and sufficient for Q” is also seen as “Q iff P.” The corresponding logical sym-
bols for “if and only if” are )and .xalim"xalim"