Bonus problem Let φ:R→R be a function. Prove that φ is convex in the sense of Exercise 1 if and only if, for any x,y∈R and θ∈[0,1], we have φ(θx+(1−θ)y)≤θφ(x)+(1−θ)φ(y). Show also that a convex function is continuous, possesses left and right derivative at every point, and is differentiable on the complement of a countable subsets.
Bonus problem Is it possible to construct a function φ:R→R that is differentiable outside a countable set N⊂R such that φ′(x)=0 for x∉N and φ(x)→±∞ as x→±∞ ?
Bonus problem Let φ:[0,1]→R be a differentiable function. Is it true that the derivative φ′:[0,1]→R is integrable in the sense of Riemann and φ(1)−φ(0)=∫10φ′(y)dy?
Tasks for June 29: resolve Exercises 1-5
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Let φ:R→R be a function. Prove that φ is convex in the sense of Exercise 1 if and only if, for any x,y∈R and θ∈[0,1], we have
φ(θx+(1−θ)y)≤θφ(x)+(1−θ)φ(y).
Show also that a convex function is continuous, possesses left and right derivative at every point, and is differentiable on the complement of a countable subsets.
Tasks for June 30: resolve Exercises 6-9
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Is it possible to construct a function φ:R→R that is differentiable outside a countable set N⊂R such that φ′(x)=0 for x∉N and φ(x)→±∞ as x→±∞ ?
Tasks for July 1: resolve Exercises 10-13
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Let φ:[0,1]→R be a differentiable function. Is it true that the derivative φ′:[0,1]→R is integrable in the sense of Riemann and
φ(1)−φ(0)=∫10φ′(y)dy?
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DeleteCan we say for opposing example ϕ(x)=x13?
DeleteThis function is not differentiable at x=0.
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DeleteIndeed, may be f(x)={$x2sin(1x2),x≠0$0,x=0
DeleteYes, this is a counterexample showing that the property is not true.
DeleteTasks for July 2: resolve Exercises 14-18
ReplyDeleteLet φ:[0,1]→R be a differentiable function such that the derivative φ′:[0,1]→R is bounded. Prove that
φ(x)=φ(0)+∫x0φ′(y)dy,x∈[0,1].
Tasks for July 8: resolve Exercises 22-24 and finish the bonus problem of June 29
ReplyDeleteTasks for July 9: resolve Exercises 25-27
ReplyDeleteTasks for July 12 & 13: revision of the first 27 exercises
ReplyDeleteTasks for July 14: complete the revision of the first 27 exercises
ReplyDeleteTasks for July 15: resolve Exercises 28-30 and write down their solutions
ReplyDeleteTasks for July 16: resolve Exercises 31-35 and write down their solutions
ReplyDeleteTasks for July 19: resolve Exercises 36-38 and write down their solutions
ReplyDeleteTasks for July 21: resolve Exercises 39-42 and write down their solutions
ReplyDeleteTasks for July 22: resolve Exercises 43-45 and write down their solutions
ReplyDeleteTasks for July 22: resolve Exercises 46-48 and write down their solutions
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