In lectures I came across a particular statement that got me confused.
I will give full context:
The context is the following proof: given a piecewise continously differentiable L-periodic function [imath]f[/imath], the fourier sereis of f converges to f.
During this proof it must be shown that [math]F_x(t) := \frac{f(x+t) - f(x)}{sin(\frac{\pi t}{L})}[/math] is Riemann-integrable. Here x is fixed such that f is differentiable at x.
The statement in the proof was merely:
By L'hospital's rule [imath]\lim_{t \rightarrow kL}F_x(t)[/imath] exists, where k is a whole number.
From this (and some hidden steps) it follows that [imath]F_x(t)[/imath] is integrable. But I'm having a hard time finding the missing steps.
I see that L'hospital's rule can be used in this case and that the above limit exists. But I don't understand how that connects to integrability.
My intuition for showing integrability is to define and upper und lower step function and then show that their difference is smaller than some [imath]\epsilon >0[/imath]. How else can you show that a function is Riemann-integrable? Maybe in the proof they use the fact that every continious function defined on a bounded interval is integrable? But I don't see how...
I will give full context:
The context is the following proof: given a piecewise continously differentiable L-periodic function [imath]f[/imath], the fourier sereis of f converges to f.
During this proof it must be shown that [math]F_x(t) := \frac{f(x+t) - f(x)}{sin(\frac{\pi t}{L})}[/math] is Riemann-integrable. Here x is fixed such that f is differentiable at x.
The statement in the proof was merely:
By L'hospital's rule [imath]\lim_{t \rightarrow kL}F_x(t)[/imath] exists, where k is a whole number.
From this (and some hidden steps) it follows that [imath]F_x(t)[/imath] is integrable. But I'm having a hard time finding the missing steps.
I see that L'hospital's rule can be used in this case and that the above limit exists. But I don't understand how that connects to integrability.
My intuition for showing integrability is to define and upper und lower step function and then show that their difference is smaller than some [imath]\epsilon >0[/imath]. How else can you show that a function is Riemann-integrable? Maybe in the proof they use the fact that every continious function defined on a bounded interval is integrable? But I don't see how...
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