Let denote the fractional part. Prove that

for the different values of the integer number .

**Solution**

Let denote the integral,

since if whereas if . Therefore,

except of a countable set whose measure is .

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Let denote the fractional part. Prove that

for the different values of the integer number .

**Solution**

Let denote the integral,

since if whereas if . Therefore,

except of a countable set whose measure is .