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 function f(x) be differentiable on [a,b], i.e. the derivative exists at every point of [a,b]. Is it correct that such function must be continuous and bounded on [a,b]? Prove, or find a counter-example.
2. Suppose a function f(x) has a left-hand derivative at any point of an open interval (a,b). Is it correct that such function must be continuous on (a,b)? Prove, or find a counter-example.
3. Prove that if functions f(x) and g(x) are integrable on [a,b], then the product f(x)g(x) is also integrable on [a,b].
4. Let function f(x) be differeniable on (a,b). Is it correct that such function must be integrable on [a,b]? Prove, or find a counter-example.