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# Properties of an alternate definition of the integral of a step function

Consider if we chose the following definition for the definite integral of a step function:

Which of the following properties (all valid for the actual definition) would remain valid under this new definition:

1. .
2. .
3. .
4. .
5. If for all , then .

1. True.
Proof. Let be a partition of and be a partition of such that is constant on the open subintervals of and . Then, let , where , so is a partition of and is constant on the open subintervals of . Then,

2. False.
Counterexample: Let for all . Then,

while

Hence, this property does not hold. (More generally, since .)

3. False. Again, this is because . A specific counterexample is given.
Counterexample: Let for all and let . Then,

while,

4. True.
Proof. Let be a partition of such that on the th open subinterval of . Then,

is a partition of and on . So,

Thus, indeed we have

5. True.
Proof. Since implies , the result follows immediately

### One comment

1. uno y dos says:

Wouldn’t it be P_1 = {x_0,…,x_m} partition of [a,b] ?