The function f(x) = begin{cases} sgn([x]) & x | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let us analyze the function $f(x)$ for different intervals.For $x 
otin I$,$f(x) = sgn([x])$.If $x \in (0, 1)$,$[x] = 0$,so $f(x) = sgn(0) = 0$.I

Vedclass · Accepted Answer

Discontinuous at every integer

Vedclass · Answer

(B) Let us analyze the function $f(x)$ for different intervals.For $x 
otin I$,$f(x) = sgn([x])$.If $x \in (0, 1)$,$[x] = 0$,so $f(x) = sgn(0) = 0$.If $x \in (1, 2)$,$[x] = 1$,so $f(x) = sgn(1) = 1$.If $x \in (-1, 0)$,$[x] = -1$,so $f(x) = sgn(-1) = -1$.For $x \in I$,$f(x) = [sgn(x)]$.If $x = 0$,$f(0) = [sgn(0)] = [0] = 0$.If $x > 0$ and $x \in I$,$f(x) = [sgn(x)] = [1] = 1$.If $x Combining these,the function is discontinuous at every integer $n \in I$ because the left-hand limit and right-hand limit do not match the function value at integers. Specifically,for any integer $n$,$\lim_{x 	o n^-} f(x) = sgn(n-1)$ and $\lim_{x 	o n^+} f(x) = sgn(n)$. These are not equal to $f(n) = [sgn(n)]$. Thus,the function is discontinuous at every integer.

The function $f(x) = \begin{cases} sgn([x]) & x \notin I \\ [sgn(x)] & x \in I \end{cases}$ is (where $sgn()$ denotes the signum function and $[.]$ denotes the greatest integer function):

Similar Questions

The number of discontinuities of the greatest integer function $f(x) = [x]$ for $x \in \left(-\frac{7}{2}, 100\right)$ is:

Define $f: R \rightarrow R$ by $f(x) = [x] + \sqrt{x - [x]}$ for $x \in R$,where $[x]$ denotes the greatest integer function. Then the set of points at which $f$ is continuous is

If $f(x) = \frac{\log_e(1 + x^2 \tan x)}{\sin x^3}, x \neq 0$ is to be continuous at $x = 0$,then $f(0)$ must be equal to

The function $f(x) = 2x - |x - x^2|$ is

Let $f(x) = \begin{cases} (x - 1)^{\frac{1}{2 - x}}, & x > 1, x \neq 2 \\ k, & x = 2 \end{cases}$. The value of $k$ for which $f$ is continuous at $x = 2$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module