The discussion centers on the approximation of continuous functions by differentiable ones, specifically through the function g defined as g(x) = (1/2δ) ∫ (from x-δ to x+δ) f. Participants are tasked with demonstrating that g is continuously differentiable and that if f is uniformly continuous, there exists a δ1 such that the supremum of the difference between f and g is less than ε for 0<δ≤δ1. Key points include the application of the fundamental theorem of calculus and the importance of uniform continuity in the analysis.
PREREQUISITESMathematics students, particularly those studying real analysis, as well as educators and tutors looking to deepen their understanding of function approximation and continuity concepts.
Frankly, I don't understand this part. If f is not linear, or at least piecewise-linear, g will depend upon both x and \delta. Was there a "\lim_{\delta\to 0}" in the definition?cooljosh2k2 said: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
?? How do you arrive at the conclusion that the integral is identically 1? The value of the integral will strongly depend upon 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
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.
and for part b, I am just trying to figure out part a before tackling part b.