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Approximation of continuous functions by differentiable ones

  • #1

Homework Statement


Let f: R-->R be continuous. For δ>0, define g: R-->R by:

g(x) = (1/2δ) ∫ (from x-δ to x+δ) f

Show:

a) g is continuously differentiable

b) If f is uniformly continuous, then, for every ε>0, there exists a δ1>0 such that sup{∣f(x) - g(x)∣; x∈R} < ε for 0<δ≤δ1


The Attempt at a Solution



Please help me, im confused
 
Last edited:

Answers and Replies

  • #2
HallsofIvy
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Homework Statement


Let f: R-->R be continuous. For δ>0, define g: R-->R by:

g(x) = (1/2δ) ∫ (from x-δ to x+δ) f
Frankly, I don't understand this part. If f is not linear, or at least piecewise-linear, g will depend upon both x and [itex]\delta[/itex]. Was there a "[itex]\lim_{\delta\to 0}[/itex]" in the definition?

Show:

a) g is continuously differentiable

b) If f is uniformly continuous, then, for every ε>0, there exists a δ1>0 such that sup{∣f(x) - g(x)∣; x∈R} < ε for 0<δ≤δ1


The Attempt at a Solution



For part a, according to the fundamental theorem of calculus, if i integrate the given integral, i get a value of 1. How do i show that g is continuously differentiable though.
?? How do you arrive at the conclusion that the integral is identically 1? The value of the integral will strongly depend upon f.

and for part b, im just trying to figure out part a before tackling part b.
 
  • #3
No, there was no limit defined in the problem, i typed it exactly how my prof. worded it.

And you are right about my conclusion for the integral, had a brain cramp and figured that f is equal to 1, which would make my integrating very easy. So completely disregard that i said that.
 

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