Functions and maps
Functions
In this course we define functions as objects which take an argument in and return a value in . We use the notation
to refer to a function with “input” space . The “output” space for functions is .
Example: The function with values
gives the distance from the point to .
We allow for functions to take infinity values. The domain of a function , denoted , is defined as the set of points where the function is finite.
Example: Define the logarithm function as the function , with values if , and otherwise. The domain of the function is thus (the set of positive reals).
Maps
We reserve the term map to refer to functions which return more than a single value, and use the notation
to refer to a map with input space and output space . The components of the map are the (scalar-valued) functions .
Example: A map. |
The map with values |
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has components the functions , with values |
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