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# Sorry for the repost, last time the guest gave no explanation, though it might be correct, I always look for the knowledge.

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How many numbers are in the range of the function $f(x) = \frac{\lceil 5x - 5\lfloor x \rfloor \rceil}5$?

Dec 22, 2020

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Let's pick  a few values and  see what happens

x= 1....ceiling  [ 5 (1)  - 5 floor (1) ] / 5 =   ceiling [ 5 - 5 ] / 5  =  ceiling 0   = 0

x  =1.2.....ceiling  [ 5(1.2)  - 5 floor (1.2) ]  /5 =  ceiling [ 6 - 5 ] / 5 =    1/5

x = 1.4  .....ceiling  [ 5 (1.4)  -5 floor (1.4) ] / 5 =  ceiling [ 7 - 5(1) ]/ 5 = 2/5

x = 1.6  .....ceiling [ 5 (1.6 )  - 5floor (1.6) ] / 5 = ceiling  [ 8 - 5 ] /5  =  3/5

x  =1.8.....ceiling  [ 5 (1.8) - 5 floor (1.8) ] /5  = ceiling [ 9 - 5] /5 = 4/5

x = 1.9.....ceiling  [ 5 (1.9)  - 5 floor (1.9] / 5  = ceiling [  9.5 - 5 ] / 5  = ceiling (4.5) /5 = 5/5  =1

And look at the  graph here : https://www.desmos.com/calculator/ixgipsfctk

It appears that  this patern is  repeated infinitely between integers

So....there are 6 numbers in the range of  this function

0, 1/5 , 2/5 , 3/5 , 4/5  and 1   Dec 22, 2020