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

 

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