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Unformatted text preview: CM221A ANALYSIS I NOTES ON WEEK 5 FUNCTIONS OF A REAL VARIABLE Let Ω ⊆ R be a nonempty subset of the real line. A (real-valued) function f on Ω is a mapping Ω → R , that is, an association x 7→ f ( x ) of each element x of Ω to some real number f ( x ) which is called the value of the function f at the point x . Further on, we shall usually be assuming that Ω is a nondegenerate interval (which may coincide with R ). The set Ω is called the domain of definition of the function. The set of all its values f ( x ) (when x runs over Ω) is said to be the range of f . Finally, the set of points ( x,y ) ∈ R 2 such that y = f ( x ) is called the graph of the function f . The graph can be thought of as a curve line in the two dimensional space R 2 whose intersection with every vertical straight line passing through ( x, 0) consists of one point with coordinates ( x,f ( x )). Every function is uniquely defined by its graph. Some functions are given by “nice” explicit formulae. For example, f ( x ) = x 2 + x +1 is a function on the whole real line, f ( x ) = ( √ x )- 1 is a real-valued function on the positive half-line (0 , + ∞ ), and so on. But it is not always the case. For instance, f ( x ) = ( 1 , if x is rational,...
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This document was uploaded on 01/31/2011.
- Spring '09