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Riemann integrable

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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 = 0
1/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 <= 1
0 if x = 0

Also show that F(x) is Reimann Integrable

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This solution helps explore Reimann Integrable functions within the context of real analysis.

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