Let f : R → R be continuous on R. For k E N, define fk : R → R by k f. 2 Prove that for all L > 0 we have that (fk)kEN Converges uniformly on [-L, L] to f. (Here the significance of the factor is that it equals e-+)) 1
Let f : R → R be continuous on R. For k E N, define fk : R → R by k f. 2 Prove that for all L > 0 we have that (fk)kEN Converges uniformly on [-L, L] to f. (Here the significance of the factor is that it equals e-+)) 1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please solve the question regarding fk(x), you could use (b) to help solve it if needed.
Transcribed Image Text:Let S be a closed n-cube with v(S) > 0 and suppose f : S → R is continuous on S (relative
to S).
(a)
Prove that infs f and sups f both exist, and that infs f < „S) Js f < sups f.
v(S)
(Ъ)
Prove that there exists xo E S such that f(xo) = „S) Js f.
![Let f : R → R be continuous on R. For k E N, define fk : R → R by
k
fk (x):
f.
Prove that for all L > 0 we have that (fk)kEN converges uniformly on [-L, L] to ƒ. (Here
the significance of the factor is that it equals](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1db0a8d-7a0a-4a06-9757-d3194d74bf23%2F0adb19e5-8ad8-4dca-a12f-ac42b2304a3f%2Fzqf7ule_processed.png&w=3840&q=75)
Transcribed Image Text:Let f : R → R be continuous on R. For k E N, define fk : R → R by
k
fk (x):
f.
Prove that for all L > 0 we have that (fk)kEN converges uniformly on [-L, L] to ƒ. (Here
the significance of the factor is that it equals
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