# Differential Equations. Proc. conf. Sao Paolo, 1981 by D. G. de Figueiredo, C. S. Hönig

By D. G. de Figueiredo, C. S. Hönig

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4. 8 Restrictions. Let T : E → E ′ be continuous. For D contained in E, the restriction of T to D is the mapping S : D → E ′ defined by putting Sx = T x for each x ∈ D. Obviously, the continuity of T implies that of S. 9 Discontinuity. Let f : R → R be defined by setting f (x) = 1 if x is rational and f (x) = 0 if x is irrational. This function is discontinuous at every x in R. To see this, fix x in R. For every δ > 0, the ball B(x, δ) has infinitely many rationals and infinitely 50 Functions on Metric Spaces Chap.

For instance, the function f : (0, 1) → [−1, 1] defined by f (x) = sin(1/x) is continuous but not uniformly so. 11 are uniformly continuous. 10 on Lipschitz continuity. Being Lipschitz almost encapsulates the notion of uniform continuity. 54 Functions on Metric Spaces Chap. 26 PROPOSITION. Let T : E → E ′ be Lipschitz continuous. Then T is uniformly continuous. PROOF. Fix ϵ > 0 and choose δ = ϵ/K. This δ works and is independent of x. 34 for an “almost converse” to this result. The following shows the important role of compactness in relation to uniform continuity.

Our aim is to introduce the idea of distance in general so that we can talk of the distance between two functions with the same conceptual ease as when we talk of the distance between two points in a plane. After that, we discuss the main issues: convergence, continuity, approximations. All along, there will be examples of different spaces and different ways of measuring distances. A. Euclidean Spaces This section will review the space Rn together with its Euclidean distance. Recall that each element of Rn is an n-tuple x = (x1 , .

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