Show that the function g(x) = x - [x] is discontinuous at all int… - Vidyadip

Question

Show that the function g(x) = x - [x] is discontinuous at all integral points. Here [x] denotes the greatest integer function.

✓

Answer

The given function is g(x) = x - [x]
It is evident that g is defined at all integral points.
Let n be an integer.
Then,
g(n) = n - [n] = n - n = 0
The left hand limit of f at x = n is,
$\lim\limits_{{\text{x}}\rightarrow\text{n}^-}\text{g(x)}=\lim\limits_{{\text{x}}\rightarrow\text{n}^-}\big(\text{x}-[\text{x}]\big)=\lim\limits_{{\text{x}}\rightarrow\text{n}^-}(\text{x})-\lim\limits_{{\text{x}}\rightarrow\text{n}^-}[\text{x}]\\=\text{n}-(\text{n}-1)=1$
The right hand limit of f at x = n is,
$\lim\limits_{{\text{x}}\rightarrow\text{n}^+}\text{g(x)}=\lim\limits_{{\text{x}}\rightarrow\text{n}^+}\big(\text{x}-[\text{x}]\big)\\=\lim\limits_{{\text{x}}\rightarrow\text{n}^+}(\text{x})-\lim\limits_{{\text{x}}\rightarrow\text{n}^+}[\text{x}]=\text{n}-\text{n}=0$
It is observed that the left and right hand limits of f at x = n do not coincide.
Therefore, f is not continuous at x = n
Hence, g is discontinuous at all integral points.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from Continuity→Continuity — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Solve the following differential equation
$\text{xy}(\text{y}+1)\text{dy}=(\text{x}^2+1)\text{dx}$

A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lowermost. Its semi-vertical angle is $\tan ^{-1}(0.5)$. Water is poured into it at a constant rate 5 cubic meter per minute. Find the rate at which the level of water rising at the instant when the depth of water in the tank is 10 meter.

The contents of urns I, II, III are as follows:
Urn I : 1 white, 2 black and 3 red balls
Urn II : 2 white, 1 black and 1 red balls
Urn III : 4 white, 5 black and 3 red balls.
One urn is chosen at random and two balls are drawn. They happen to be white and red. What is the probability that they come from Urns I, II, III?

Find the general solution of $\left(1+x^{2}\right) d y+2 x y d x=\cot x d x(x \neq 0)$

Show that if x² – 5x + 6 = 0, then x = 3 or x = 2

If $\text{A}=\begin{bmatrix}1&2&0\\3&-4&5\\0&-1&3\end{bmatrix},$ compute A² - 4A + 3I₃.

Find the general solution of the differential equation $\text{x}\frac{\text{dy}}{\text{dx}}+2{\text{y}}=\text{x}^2$

Find the intervals in which the following functions are increasing or decreasing.
f(x) = 5 + 36x + 3x²- 2x³

Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}\text{a}&\text{b}&\text{c}\\\text{a}+2\text{x}&\text{b}+2\text{y}&\text{c}+2\text{z}\\\text{x}&\text{y}&\text{z}\\\end{vmatrix}$

Let $\text{F}(\alpha)=\begin{bmatrix}\cos\alpha & -\sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1\end{bmatrix}$

$\text{and G }(\beta)=\begin{bmatrix} \cos\beta & 0 & \sin\beta \\ 0 & 1 & 0 \\ -\sin\beta & 0 & \cos\beta \end{bmatrix}$

$\big[\text{F}(\alpha)\text{G}(\beta)\big]^{-1}=\text{G}(-\beta)\text{F}(-\alpha).$