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Suppose f is a real continuous function on R', f.(t)=f(nt) for n=1, 2, 3, ..., and {S} is equicontinuous on [0, 1]. What conclusion can you draw about f?

Suppose f is a real continuous function on R', f.(t)=f(nt) for n=1, 2, 3, ..., and {S} is equicontinuous on [0, 1]. What conclusion can you draw about f?

College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...
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168 PRINCIPLES OF MATHEMATICAL ANALYSIS 14. Let f be a continuous real function on R' with the following properties: Os f(1)<1, f(t+ 2) = f(t) for every t, and (0 f(t)= 11 (SIS1). Put 9(t) = (x(t), y(t)), where x(1) =E2-"f(32n - 1f), -E2-(3"). y(t) = Prove that 0 is continuous and that O maps I= [0, 1] onto the unit square 12 c R?. If fact, show that O maps the Cantor set onto 1?. Hint: Each (xo, Yo) e 1? has the form Xo = Yo = 2-"azn where each a, is 0 or 1. If to = ' 3 -1-'(2a,) show that f(3*to) = ar, and hence that x(to) = xo, y(to) = yo. (This simple example of a so-called "space-filling curve" is due to I. J. Schoenberg, Bull. A.M.S., vol. 44, 1938, pp. 519.) 15. Suppose f is a real continuous function on R', f.(1)=f(nt) for n=1, 2, 3, ..., and {f) is equicontinuous on [0, 1]. What conclusion can you draw about f? 16. Suppose {f.} is an equicontinuous sequence of functions on a compact set K, and {f} converges pointwise on K. Prove that {fa} converges uniformly on K. 17. Define the notions of uniform convergence and equicontinuity for mappings into any metric space. Show that Theorems 7.9 and 7.12 are valid for mappings into any metric space, that Theorems 7.8 and 7.11 are valid for mappings into any complete metric space, and that Theorems 7.10, 7.16, 7.17, 7.24, and 7.25 hold for vector-valued functions, that is, for mappings into any R*. 18. Let {f.} be a uniformly bounded sequence of functions which are Riemann-inte- grable on [a, b], and put F.(x) = fa(t) (a<xsb). Prove that there exists a subsequence {F} which converges uniformly on [a, b]. 19. Let K be a compact metric space, let S be a subset of C(K). Prove that S is compact (with respect to the metric defined in Section 7.14) if and only if S is uniformly closed, pointwise bounded, and equicontinuous. (If S is not equicontinuous, then S contains a sequence which has no equicontinuous subsequence, hence has no subsequence that converges uniformly on K.) SEQUENCES AND SERIES OF FUNCTIONS 169 20. If f is continuous on [0, 1] and if (n= 0, 1, 2, ...), prove that f(x) = 0 on [0, 1]. Hint: The integral of the product of f with any
Transcribed Image Text:168 PRINCIPLES OF MATHEMATICAL ANALYSIS 14. Let f be a continuous real function on R' with the following properties: Os f(1)<1, f(t+ 2) = f(t) for every t, and (0 f(t)= 11 (SIS1). Put 9(t) = (x(t), y(t)), where x(1) =E2-"f(32n - 1f), -E2-(3"). y(t) = Prove that 0 is continuous and that O maps I= [0, 1] onto the unit square 12 c R?. If fact, show that O maps the Cantor set onto 1?. Hint: Each (xo, Yo) e 1? has the form Xo = Yo = 2-"azn where each a, is 0 or 1. If to = ' 3 -1-'(2a,) show that f(3*to) = ar, and hence that x(to) = xo, y(to) = yo. (This simple example of a so-called "space-filling curve" is due to I. J. Schoenberg, Bull. A.M.S., vol. 44, 1938, pp. 519.) 15. Suppose f is a real continuous function on R', f.(1)=f(nt) for n=1, 2, 3, ..., and {f) is equicontinuous on [0, 1]. What conclusion can you draw about f? 16. Suppose {f.} is an equicontinuous sequence of functions on a compact set K, and {f} converges pointwise on K. Prove that {fa} converges uniformly on K. 17. Define the notions of uniform convergence and equicontinuity for mappings into any metric space. Show that Theorems 7.9 and 7.12 are valid for mappings into any metric space, that Theorems 7.8 and 7.11 are valid for mappings into any complete metric space, and that Theorems 7.10, 7.16, 7.17, 7.24, and 7.25 hold for vector-valued functions, that is, for mappings into any R*. 18. Let {f.} be a uniformly bounded sequence of functions which are Riemann-inte- grable on [a, b], and put F.(x) = fa(t) (a<xsb). Prove that there exists a subsequence {F} which converges uniformly on [a, b]. 19. Let K be a compact metric space, let S be a subset of C(K). Prove that S is compact (with respect to the metric defined in Section 7.14) if and only if S is uniformly closed, pointwise bounded, and equicontinuous. (If S is not equicontinuous, then S contains a sequence which has no equicontinuous subsequence, hence has no subsequence that converges uniformly on K.) SEQUENCES AND SERIES OF FUNCTIONS 169 20. If f is continuous on [0, 1] and if (n= 0, 1, 2, ...), prove that f(x) = 0 on [0, 1]. Hint: The integral of the product of f with any
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