Continuity

Question

Let $f(x) = [x]^2 + \sqrt{\{x\}}$, where $[.]$ denotes the greatest integer function and $\{.\}$ denotes the fractional part. Discuss the continuity of $f(x)$ at integer points $x = n$.

RankGuru · Accepted Answer

We analyze the continuity at an integer $n$.

1. Right Hand Limit (RHL) at $x 	o n^+$:
$[x] = n, \{x\} 	o 0$. So, RHL $= n^2 + \sqrt{0} = n^2$.

2. Left Hand Limit (LHL) at $x 	o n^-$:
$[x] = n - 1, \{x\} 	o 1$. So, LHL $= (n-1)^2 + \sqrt{1} = (n-1)^2 + 1$.

3. Condition for continuity:
RHL $= $ LHL $\implies n^2 = (n-1)^2 + 1$
$\implies n^2 = n^2 - 2n + 1 + 1$
$\implies 0 = -2n + 2$
$\implies 2n = 2 \implies n = 1$.

4. Verification at $n=1$:
$f(1) = 1^2 + 0 = 1$. RHL $= 1$. LHL $= 0^2 + 1 = 1$. Since LHL $=$ RHL $= f(1)$, $f(x)$ is continuous at $x=1$. For all other integers, it is discontinuous.

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