If function f(x) = begin{cases} x, & text{if | vedclass

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

Question

(B) For $f(x)$ to be continuous at a point $x = a$,the limit $\lim_{x 	o a} f(x)$ must exist and be equal to $f(a)$.Consider the point $x = a$. For a

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(B) For $f(x)$ to be continuous at a point $x = a$,the limit $\lim_{x 	o a} f(x)$ must exist and be equal to $f(a)$.Consider the point $x = a$. For any neighborhood of $a$,there exist both rational and irrational numbers.If $x$ is rational,$f(x) = x$,and if $x$ is irrational,$f(x) = 1 - x$.For the limit to exist at $x = a$,we must have $a = 1 - a$,which implies $2a = 1$,or $a = 1/2$.Now,check continuity at $a = 1/2$:$f(1/2) = 1/2$ (since $1/2$ is rational).However,in any neighborhood of $1/2$,there are irrational numbers $x$ such that $f(x) = 1 - x$,which approaches $1 - 1/2 = 1/2$.Wait,let us re-evaluate: $\lim_{x 	o 1/2} f(x)$ exists if the values of $f(x)$ for rational and irrational numbers approach the same value.For rational $x$,$f(x) = x 	o 1/2$.For irrational $x$,$f(x) = 1 - x 	o 1 - 1/2 = 1/2$.Since both limits are equal to $1/2$,the limit exists at $x = 1/2$.Thus,$f(x)$ is continuous only at $x = 1/2$.Therefore,the number of points of continuity is $1$.

If function $f(x) = \begin{cases} x, & \text{if } x \text{ is rational} \\ 1 - x, & \text{if } x \text{ is irrational} \end{cases}$,then $f(x)$ is continuous at how many points?

Similar Questions

Which of the following function$(s)$ not defined at $x = 0$ has/have an irremovable discontinuity at $x = 0$?

The function $f(x) = \frac{1 - \sin x + \cos x}{1 + \sin x + \cos x}$ is not defined at $x = \pi$. The value of $f(\pi)$,so that $f(x)$ is continuous at $x = \pi$,is

Let $f(x) = \begin{cases} \frac{\tan^2 \{x\}}{x^2 - [x]^2} & \text{for } x > 0 \\ 1 & \text{for } x = 0 \\ \sqrt{\{x\} \cot \{x\}} & \text{for } x < 0 \end{cases}$ where $[x]$ is the greatest integer function and $\{x\}$ is the fractional part function of $x$,then:

If $f(x) = \begin{cases} \frac{\log_{e} x}{x-1} & x \neq 1 \\ k & x=1 \end{cases}$ is continuous at $x=1$,then the value of $k$ is

Prove that every rational function is continuous at every point in its domain.

Vedclass Test Series

Exam Paper Generator

Online Exam Module