Let $f ( x ) = \lfloor x \rfloor$ Which of the following is true?

A)ƒ is undefined at 0.
B)ƒ is undefined at $- \frac { 1 } { 2 }$
C)ƒ has a removable discontinuity at 0.
D)ƒ has a removable discontinuity at $\frac { 1 } { 2 }$
E)ƒ has a jump discontinuity at 0.

Question

Let $f ( x ) = \lfloor x \rfloor$ Which of the following is true?

A)ƒ is undefined at 0.
B)ƒ is undefined at $- \frac { 1 } { 2 }$
C)ƒ has a removable discontinuity at 0.
D)ƒ has a removable discontinuity at $\frac { 1 } { 2 }$
E)ƒ has a jump discontinuity at 0.

Quizplus · Accepted Answer

The function $f(x) = \lfloor x \rfloor$ is defined for all real numbers, including 0 and $-\frac{1}{2}$, so A and B are incorrect. The function does not have a removable discontinuity at 0 or $\frac{1}{2}$ because the function value jumps from one integer to the next at every integer value, including 0. Therefore, C and D are incorrect. The correct answer is E because $f(x) = \lfloor x \rfloor$ has a jump discontinuity at 0 (and at every integer), where the function value jumps from $\lfloor -0.01 \rfloor = -1$ to $\lfloor 0 \rfloor = 0$.

Let f(x)=⌊x⌋f ( x ) = \lfloor x \rfloorf(x)=⌊x⌋ Which of the Following Is True? A)ƒ Is Undefined

Let $f ( x ) = \lfloor x \rfloor$ Which of the Following Is True?

A)ƒ Is Undefined