Question

If $f\left(x\right)=\left[x\right]-\left[\frac{x}{4}\right],x\in R$, where $\left[x\right]$ denotes the greatest integer function, then:

• A. Both ${lim}_{x\to 4-}f\left(x\right)$ and ${lim}_{x\to 4+}f\left(x\right)$ exist but are not equal
• B. ${lim}_{x\to 4-}f\left(x\right)$ exists but ${lim}_{x\to 4+}f\left(x\right)$ does not exist
• C. ${lim}_{x\to 4+}f\left(x\right)$ exist but ${lim}_{x\to 4-}f\left(x\right)$ does not exist
• D. $f$ is continuous at $x=4$

Answer 1

Answer: (D)

$f$ is continuous at $x=4$

limx→4+f(x)=limx→4+(([x]−[x4]))=4−1=3limx→4+f(x)=limx→4+(([x]−[x4]))=4−1=3

limx→4−f(x)=limx→4−(([x]−[x4]))=3−0=3limx→4−f(x)=limx→4−(([x]−[x4]))=3−0=3

f(x)=3f(x)=3

∴∴ continuous at x=4x=4