Show that the function defined by g(x)=x-[x] is discontinuous at all integral points.
Here [x] denotes the greatest integer less than or equal to x.
Answer
364.8k+ views
Hint-To solve these types of problems calculate the value of LHL and RHL and show
that the value of $LHL \ne RHL$which means to say that they are discontinuous.
The given function is g(x)=x-[x]
In this function let us consider an integer n and solve it
On substituting the value of n in the equation, we get
g(n)=n-[n]=n-n=0
Now let us take the LHL and RHL of this equation,
We get LHL at x=n=$\mathop {\lim }\limits_{x \to {n^ - }} g(x) = \mathop {\lim }\limits_{x \to
{n^ - }} (x - [x]) = n - (n - 1) = 1$
RHL at x=n=$\mathop {\lim }\limits_{x \to {n^ + }} g(x) = \mathop {\lim }\limits_{x \to
{n^ + }} (x - [x]) = n - n = 0$
So, from this we can clearly observe that the value of $LHL \ne RHL$
If, for a function $LHL \ne RHL$, then we can say that the function is discontinuous
So, we can say that g(x)=x-[x] is discontinuous at all integral points
Note: If a similar type of question is asked to show that the functions are continuous then
show that LHL=RHL , which means to say that the function is continuous.
that the value of $LHL \ne RHL$which means to say that they are discontinuous.
The given function is g(x)=x-[x]
In this function let us consider an integer n and solve it
On substituting the value of n in the equation, we get
g(n)=n-[n]=n-n=0
Now let us take the LHL and RHL of this equation,
We get LHL at x=n=$\mathop {\lim }\limits_{x \to {n^ - }} g(x) = \mathop {\lim }\limits_{x \to
{n^ - }} (x - [x]) = n - (n - 1) = 1$
RHL at x=n=$\mathop {\lim }\limits_{x \to {n^ + }} g(x) = \mathop {\lim }\limits_{x \to
{n^ + }} (x - [x]) = n - n = 0$
So, from this we can clearly observe that the value of $LHL \ne RHL$
If, for a function $LHL \ne RHL$, then we can say that the function is discontinuous
So, we can say that g(x)=x-[x] is discontinuous at all integral points
Note: If a similar type of question is asked to show that the functions are continuous then
show that LHL=RHL , which means to say that the function is continuous.
Last updated date: 28th Sep 2023
•
Total views: 364.8k
•
Views today: 5.64k
Recently Updated Pages
What do you mean by public facilities

Difference between hardware and software

Disadvantages of Advertising

10 Advantages and Disadvantages of Plastic

What do you mean by Endemic Species

What is the Botanical Name of Dog , Cat , Turmeric , Mushroom , Palm

Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Summary of the poem Where the Mind is Without Fear class 8 english CBSE

The poet says Beauty is heard in Can you hear beauty class 6 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

What is the past tense of read class 10 english CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
