Please, provide complete and detailed proofs of every statement, or give a reference to the book if you use a statement from the book.

1. Let a real-valued function f(x) be differentiable on (a,b), i.e. the derivative exists at every point of (a,b). Is it correct that the function f(x)-[f(x)], i.e. the fractional part of f(x), must be piecewise continuous on (a,b)? Prove, or find a counter-example.
2. Prove that function f(x)=1/x is differentiable at the point x=1, using the Cauchy convergence criterion for the limit.
3. Prove that, if functions f(x) and g(x) are integrable on [a,b], then the ratio f(x)/g(x) is also integrable on [a,b] provided it is bounded, or find a counter-example.
4. Let function f(x) be integrable on [a,b]. Is it correct that the function [f(x)], i.e. the integral part of f(x), must be integrable on [a,b]? Prove, or find a counter-example.