MAΤΗ Μ413 DANIELE ROSSO HW 12 (1) Prove that the function f(r) = is not uniformly contimuous on (0, 1). (Hint: find two sequences (r,}nEN and {žn}n€N such that |r,n – žn| < but |f(xn) – f(2n)| 2 1 for all n.) (2) Prove that the function f(r) = VT is uniformly continuous on [-1, ). (Hint: follow the example of f(r) = V from class, but considering the sets (-1, 1] and (1/2, ox)) (3) Prove that the piece-wise function defined by 3r if r < -2 f(z) = {-4 if a = -2
MAΤΗ Μ413 DANIELE ROSSO HW 12 (1) Prove that the function f(r) = is not uniformly contimuous on (0, 1). (Hint: find two sequences (r,}nEN and {žn}n€N such that |r,n – žn| < but |f(xn) – f(2n)| 2 1 for all n.) (2) Prove that the function f(r) = VT is uniformly continuous on [-1, ). (Hint: follow the example of f(r) = V from class, but considering the sets (-1, 1] and (1/2, ox)) (3) Prove that the piece-wise function defined by 3r if r < -2 f(z) = {-4 if a = -2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![11:54 1
< Ваck
hw12-M413.pdf Q
МАТH M413
DANIELE ROSSO
HW 12
(1) Prove that the function f(r) = is not uniformly continuous on (0, 1].
(Hint: find two sequences (a,}nEN and {2,}nEN such that ,- zn| < 1 but |f(x,) – f (2,)| 21
for all n.)
(2) Prove that the function f(x) = VE is uniformly continuous on [-1, o0).
(Hint: follow the example of f(r) = Va from class, but considering the sets [-1, 1] and [1/2, ) )
(3) Prove that the piece-wise function defined by
1
3z -
if r < -2
f(1) = {-4
if r = -2
|1² + 5x + }
if a > -2
is integrable on [-4,7).
(Hint: do not use upper and lower sums, use a theorem)
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Transcribed Image Text:11:54 1
< Ваck
hw12-M413.pdf Q
МАТH M413
DANIELE ROSSO
HW 12
(1) Prove that the function f(r) = is not uniformly continuous on (0, 1].
(Hint: find two sequences (a,}nEN and {2,}nEN such that ,- zn| < 1 but |f(x,) – f (2,)| 21
for all n.)
(2) Prove that the function f(x) = VE is uniformly continuous on [-1, o0).
(Hint: follow the example of f(r) = Va from class, but considering the sets [-1, 1] and [1/2, ) )
(3) Prove that the piece-wise function defined by
1
3z -
if r < -2
f(1) = {-4
if r = -2
|1² + 5x + }
if a > -2
is integrable on [-4,7).
(Hint: do not use upper and lower sums, use a theorem)
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