To analyze the discontinuities of the function [tex]\( f(x) = 4\lfloor x - 3 \rfloor + 2 \)[/tex], we need to understand the behavior of the floor function, [tex]\(\lfloor x - 3 \rfloor\)[/tex].
The floor function, [tex]\(\lfloor y \rfloor\)[/tex], returns the greatest integer less than or equal to [tex]\( y \)[/tex]. Hence, [tex]\(\lfloor x - 3 \rfloor\)[/tex] is the greatest integer less than or equal to [tex]\( x - 3 \)[/tex].
The floor function, [tex]\(\lfloor x - 3 \rfloor\)[/tex], is discontinuous at all points where [tex]\( x - 3 \)[/tex] is an integer because at these points, the value of [tex]\( \lfloor x - 3 \rfloor \)[/tex] jumps from one integer to another.
For [tex]\( \lfloor x - 3 \rfloor \)[/tex] to be an integer, [tex]\( x - 3 \)[/tex] must be an integer. Therefore, [tex]\( x \)[/tex] itself must be an integer because adding 3 to any integer is another integer.
Thus, the function [tex]\( f(x) = 4\lfloor x - 3 \rfloor + 2 \)[/tex] is discontinuous at all integer values of [tex]\( x \)[/tex].
Therefore, the graph of [tex]\( f(x) = 4\lfloor x - 3 \rfloor + 2 \)[/tex] is discontinuous at:
[tex]\[ \boxed{\text{all integers}} \][/tex]
