For a real number , let [ ] denote the greatest integer less than or equal to . For a finite set S, let |S| denote the number of elements in the set S. Consider the functions f:(-3,3) (- , ) and g:(-3,3) (- , ) defined by f(x)=[x3] e (1+ 2 ( (x-[x]) ) ) and g(x)=x3 2 ( e(1+x-[x]) ). Let A=\x (-3,3): f is discontinuous at x\ and B=\x (-3,3): g is discontinuous at x\. Then the value of |A|+2|B|-|A B| is .
Answer
56
Solution
2( (x-[x]))= 2( x), so h(x)= e(1+ 2 x) is continuous and h=0 only at integers.
[x3] jumps where x3=k, k=-26, ,26: 53 points. At x=0, 1, 2 (x3 integer and x integer) h=0, so f is continuous there.
|A|=53-5=48 (none of them is an integer).
For g: at an integer n, right limit =n3 20=0, left limit =n3 2( e2) 0 unless n=0. Elsewhere g is continuous.
B=\-2,-1,1,2\, |B|=4, A B= .
|A|+2|B|-|A B|=48+8-0=56.
Answer: 56