Ceiling and floor functions: IITJEE mains training

Problem 1:

For what values of x, is (a) $\lfloor x \rfloor =0$ (b) $\lceil x \rceil =0$ ?

Problem 2:

Which real numbers x satisfy the equation $\lfloor x \rfloor = \lceil x \rceil$ ?

Problem 3:

Does $\lceil (-x) \rceil = - (\lfloor x \rfloor)$ for all real x? Give reasons for your answer.

Problem 4:

Graph: $f(x)=\lfloor x \rfloor$ when $x \geq 0$ ; and $f(x) = \lceil x \rceil$ , when $x <0$ .

Why is f(x) called the integer part of x? Discuss the continuity and differentiability of f(x).

Cheers,

Nalin Pithwa

Mathematics Hothouse