By Anthony C. Grove

This textbook introduces the recommendations and functions of either the laplace rework and the z-transform to undergraduate and training engineers. the expansion in computing strength has intended that discrete arithmetic and the z-transform became more and more vital. The textual content contains the mandatory concept, whereas keeping off an excessive amount of mathematical aspect, makes use of end-of-chapter routines with solutions to stress the thoughts, good points labored examples in each one bankruptcy and gives commonplace engineering examples to demonstrate the textual content.

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**Example text**

It is obvious (by temporarily fixing one of x1 , x 2) that uniform continuity implies continuity at every given point; thus uniform continuity implies continuity. A standard counterexample showing that the converse fails is the function f ( x ) = x1 from the interval (0,1) (considered as a metric subspace of ») into ». The conditions under which continuity implies uniform continuity will be studied later. Continuous mappings do not always carry a Cauchy sequence into a Cauchy sequence (Exercise 7).

Similarly, d(a, b) ≤ d(a, xi ) + d( xi , yi ) + d( yi , b) ≤ d( xi , yi ) + ε, for all i > N . It follows that, for all i > N , d(a, b) − ε ≤ d( xi , yi ) ≤ d(a, b) + ε. All this tells us that the sequence (d( xi , yi )) converges to the number d(a,b). indd 27 2/6/15 5:10 PM 28 ◾ Metric Spaces: Definition, Examples, and Basics It follows from Proposition 3 that d, considered as a mapping X 2 → », where X 2 is ☐ equipped with the product metric d p , is always continuous. We now strengthen the definition of continuity.

We leave the justification as an exercise (Exercise 19). A dilation with 0 < α < 1 is called a contraction by a fixed factor α. If there is α ∈(0, 1) such that d2 ( f ( x1 ), f ( x 2 )) ≤ αd1 ( x1 , x 2 ) for every x1 , x 2 ∈ X1 , then we say that f is a contraction. It is very easy to see that contractions must be continuous. A mapping f : X → X has a fixed point if there is an element x ∈ X such that f ( x ) = x . Theorem 6 (Banach fixed point theorem). Let X be a complete metric space. Every contraction X → X has a unique fixed point.