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 "Solve the Following Question.(5 Marks)" from Continuity→Continuity — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find the area of the ragion in the first quadrant bounded by the parabola $y=4 x^2$ and the lines $x=0, y=1$ and $y=4$.

Find the value of $\lambda$ for which the four points with position vectors
$-\hat{\text{j}}-\hat{\text{k}},4\hat{\text{i}}+5\hat{\text{j}}+\lambda\hat{\text{k}},3\hat{\text{i}}+9\hat{\text{j}}+4\hat{\text{k}}$ and $-4\hat{\text{i}}+4\hat{\text{j}}+4\hat{\text{k}}$ are co planar.

Find gof and fog when $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by: $f(x)=2 x+3$ and $g(x)=x^2+5$

A firm manufactures $3$ products A, B and C. The profits are $Rs. 3, Rs. 2$ and $Rs. 4$ respectively. The firm has $2$ machines and below is the required processing time in minutes for each machine on each product:

Machine	Products
Machine	A	B	C
$M_1$	$4$	$3$	$5$
$M_2$	$2$	$2$	$4$

Machines $M_1$ and $M_2$ have $2000$ and $2500$ machine minutes respectively. The firm must manufacture $100 A's$, $200 B's$ and $50 C's$ but not more than $150 A's$. Set up a LPP to maximize the profit.

Differentiate the following functions with respect to x:
$\cos^{-1}\Big\{\sqrt{\frac{1+\text{x}}{2}}\Big\},-1<\text{x}<1$

An anti-aircraft gun can take a maximum of 4 shots at an enemy plane moving away from it. The probabilities of hitting the plane at the first, second, third and fourth shot are 0.4, 0.3, 0.2 and 0.1 respectively. What is the probability that the gun hits the plane?

Find the area of the region bounded by the curve $(y-1)^2=4(x+1)$ and the line $y=(x-$

1).

If $\text{A}=\begin{bmatrix}2&-3&-5\\-1&4&5\\1&-3&-4\end{bmatrix}$ and $\text{B}=\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix},$ show that AB = A and BA = B.

If f'(x) = a sin x + b cos x and f'(0) = 4, f(0) = 3, $\text{f}\Big(\frac{\pi}{2}\Big)=5$, find f(x).

A company has two plants to manufacture bicycles. The first plant manufactures $60 \%$ of the bicycles and the second plant $40 \%$. Out of the $80 \%$ of the bicycles are rated of standard quality at the first plant and $90 \%$ of standard quality at the second plant. A bicycle is picked up at random and found to be standard quality. Find the probability that it comes from the second plant.