Integration of a function

(a) let f:[0,1] ---R be the function

f(x) = { x when x is an element of rational numbers
{-x when x is not an element of rational numbers

Prove that f is not integrable on [0,1] but |f| is integrable

(b) Find the limit as x goes to 0 of
1/x the integral of e^t^2 dt between the boundaries 0 and x, x being the upper boundary.

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...|f| is integrable

(b) Find the limit as x goes to 0 of
1/x the integral of e^t^2 dt between the boundaries 0 and x, x being the upper boundary.

Solution:

(a) Clearly f(x) is not continuous on [0,1], thus it ...