discuss in detail continuity of floor and ceiling functions.
Answers
In mathematics and computer science, the floor function is the function that takes as input a real number {\displaystyle x}x, and gives as output the greatest integer less than or equal to {\displaystyle x}x, denoted {\displaystyle \operatorname {floor} (x)}{\displaystyle \operatorname {floor} (x)} or {\displaystyle \lfloor x\rfloor } \lfloor x\rfloor. Similarly, the ceiling function maps {\displaystyle x}x to the least integer greater than or equal to {\displaystyle x}x, denoted {\displaystyle \operatorname {ceil} (x)}{\displaystyle \operatorname {ceil} (x)} or {\displaystyle \lceil x\rceil }{\displaystyle \lceil x\rceil }.[1]
For example, {\displaystyle \operatorname {floor} (2.4)=\lfloor 2.4\rfloor =2}{\displaystyle \operatorname {floor} (2.4)=\lfloor 2.4\rfloor =2} and {\displaystyle \operatorname {ceil} (2.4)=\lceil 2.4\rceil =3}{\displaystyle \operatorname {ceil} (2.4)=\lceil 2.4\rceil =3}, while {\displaystyle \lfloor 2\rfloor =\lceil 2\rceil =2}{\displaystyle \lfloor 2\rfloor =\lceil 2\rceil =2}.
The integral part or integer part of x, often denoted {\displaystyle [x],}{\displaystyle [x],} is {\displaystyle \lfloor x\rfloor }\lfloor x\rfloor if x is nonnegative, and {\displaystyle \lceil x\rceil }\lceil x\rceil otherwise. In words, this is the integer that has the largest absolute value less than or equal to the absolute value of x.