Question

find all the critical points of the greatest integer function f(x)=[x]

Answer 1

integers are critical points because it is not continuous at these points.
\[Let x=a, \lim_{x \rightarrow a}\left[ x \right] =\left[ a \right],if x is \not an integer ,\therefore \left[ x \right] is continuous.\]
if x is an integer,then
\[\lim_{x \rightarrow a-}\left[ x \right]=a-1,\]
\[\lim_{x \rightarrow a+}\left[ x \right]=a\]
\[hence L.H.L neqR.H.L\]
\[\left[ x \right] is \not continuous at integers.\]

Answer 2

I don't really understand your way of writing. but the answer is all real numbers
because the function looks like this:
so there are only two case
one is the function has horizontal lines
two is the function doesn not exist at integers
that means that the derivative of this function is always equal to 0 or undefined
so all real numbers are critical numbers

Answer 3

The "function" exists at the integers; the derivative does not. In other words, if \(f(x)=\lfloor x\rfloor\), then \(f(1)=1\) and \(f'(1)\) does not exist.

Answer 4

I'm sorry, I should have said the limit of the function