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Are these functions Reimann Integrable? I am just learning this topic, so my description may not be accurate. A function is Reimann Integrable if it's Upper Darboux Sums and Lower Darboux suns are equal.
Or stated another way, if U(f, P) - L(f, P) < e
The two functions are piecewise functions.
1) f(x) = { 0 when x = 01/n when x is in (1/(n+1) , 1/n]}
Show that F(x) is Reimann Integrable on [0,1]
2) f(x) = { sin(PI/x) if 0 < x <= 10 if x = 0}
Also show that F(x) is Reimann Integrable
Suppose that f_n(0) converges to some number, denoted f(0), and also suppose that the sequence (f'_n) converges uniformly on (0,1) to some function g: (0,1) -> R. Prove that the sequence (f_n) converges uniformly. What is its limit?
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