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This is the elementary real analysis assignment 5, last time I decided to tip you 20% which is $4, so is that ok

this time be $24?

There is still tips for this assignment.

MATH 3A03 Assignment #5 Due: Thursday, November 19, in class 1. Let f ( x ) be the function defined on the interval [0 , 2] such that f ( x ) = 1 for 0 x 1 and f ( x ) = 2 for 1 < x 2. Using the definition of the Riemann Integral, show that f is Riemann Integrable over the interval [0 , 2]. 2. Let f ( x ) = 1 /x for 1 x 3 and let π be the partition { 1 , 2 , 3 } of [1 , 3]. Find m ( π ) and M ( π ), the lower and upper sums for f using π , respectively. 3. Let f ( x ) be the function with domain [0 , 1] defined by f ( x ) = x if x is rational and f ( x ) = 0 if x is irrational. Determine if f is Riemann Integrable. Justify your answer. 4. Let f be continuous on the interval [ a,b ]. Show that there is some number c ( a,b ) with Z b a f ( x ) dx = f ( c )( b - a ). 5. Let g ( x ) be a continuous function on the interval [0 , 1] such that g ( x ) 0 for all x [0 , 1]. Show that if Z 1 0 g ( x ) dx = 0 then g ( x ) = 0 for all x [0 , 1]. Is this conclusion still valid if g is Riemann Integrable on [0 , 1] but it is not assumed to be continuous on [0 , 1]? Justify your answer. 1
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