Exercise 4. (i) Suppose that u :]0, 1] → R is a bounded function. Define a function v: [0, 1] → R by v(0) = 0 and, for all x €]0, 1], v(x) = x²u(x). Show that v is differentiable in 0. (ii) Find a function v: [0, 1] → R such that v is differentiable on [0, 1] and v' is unbounded on [0, 1]. You may want to apply the preceding construction to a rapidly oscillating function u.

Exercise 4. (i) Suppose that u :]0, 1] → R is a bounded function. Define a function v: [0, 1] → R by v(0) = 0 and, for all x €]0, 1], v(x) = x²u(x). Show that v is differentiable in 0. (ii) Find a function v: [0, 1] → R such that v is differentiable on [0, 1] and v' is unbounded on [0, 1]. You may want to apply the preceding construction to a rapidly oscillating function u.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.4: Spanning Sets And Linear Independence

Problem 76E: Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are...

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