Real valued multivariable function
A function with more than 1 inputs or outputs.
denotes a n-dimensional space with each axis denoted by real
numbers.
A function is a map between a ordered
n-tuple to real numbers. The domain can also be . The
same can be applied to the output of the function.
Vector-valued functions
Functions with more than 1 outputs. Referred to as vector-valued functions, even
though they are also multivariable functions.
Definitions
Suppose and .
epsilon-disk
An -disk around is the set of all points
where the distance between the points is less than .
Interior point
is an interior point of iff there exists some -disk around
that is contained in .
Boundary point
is an boundary point of iff all -disks around contains
points from and not from .
Isolated point
is an isolated point of iff there exists some -disk around
that doesn’t contains no other points of . Subset of boundary points.
Open subset
is an open subset of iff all the points of are
interior points of .
Closed subset
is an closed subset of iff contains all of its
boundary points.
Closure
The set of boundary points of and the region . Denoted by .
Bounded subset
is a bounded subset of iff is contained in some
-disk around some point.
Circular neighbourhood
The below set is called a -neighbourhood of the point and
.
Square neighbourhood
The below set is called a square-neighbourhood of the point and
.