Answer

Verified

408.6k+ views

**Hint:**Piecewise function is a function that behaves differently based on the input ‘x’ values. We can solve the given problem by taking an example. We know that a function is said to be discontinuous at ‘a’ then \[\mathop {\lim }\limits_{x \to a} f(x) \ne f(a)\]. We can also say that if the limit does not exist then we can say that it is discontinuity.Here we need to find both the left hand limit and right hand limit.

**Complete step by step answer:**

Let’s take an example.Consider,

\[f(x) = \left\{

{x^2}{\text{ }}if{\text{ }}x < 1 \\

x{\text{ }}if{\text{ 1}} \leqslant x < 2 \\

2x - 1{\text{ }}if{\text{ 2}} \leqslant x \\

\right.\]

Thus we have taken a piecewise function in which the function defined at ‘x’ values 1 and 2.

Let’s check that if ‘f’ is continuous or discontinuous at \[x = 1\].

\[\mathop {\lim }\limits_{x \to {1^ - }} f(x) = \mathop {\lim }\limits_{x \to {1^ - }} {x^2} = {1^2} = 1\]

(Take the function which is defined at \[x < 1\])

\[\mathop {\lim }\limits_{x \to {1^ + }} f(x) = \mathop {\lim }\limits_{x \to {1^ + }} x = 1\]

(Take the function which is defined at \[1 \leqslant x\])

Since both limits give the same answer,

\[\mathop {\lim }\limits_{x \to 1} f(x) = 1{\text{ }} - - - - - (1)\].

\[f(1) = 1{\text{ }} - - - (2)\]

(Take the function which is defined at \[x = 1\])

From equation (1) and (2) we have,

\[\mathop {\lim }\limits_{x \to a} f(x) = f(a)\].

Hence at \[x = 1\] the given function is not discontinuous.

Let’s check that if ‘f’ is continuous or discontinuous at \[x = 2\].

\[\mathop {\lim }\limits_{x \to {2^ - }} f(x) = \mathop {\lim }\limits_{x \to {2^ - }} x = 2\]

(Take the function which is defined at \[x < 2\])

\[\mathop {\lim }\limits_{x \to {2^ + }} f(x) = \mathop {\lim }\limits_{x \to {2^ + }} (2x - 1) = (2(2) - 1) = 3\]

(Take the function which is defined at \[2 \leqslant x\])

We can see that both give different values hence, \[\mathop {\lim }\limits_{x \to 2} f(x)\] does not exist.

**Hence, there is a jump discontinuity at \[x = 2\].**

**Note:**We know that if both right hand limit and left hand limit exist, then only the limit exists. If the limit does not exist then no need to find f(a) value. We can say that it is discontinuous. We have three different kinds of discontinuity. In the given problem we have the left hand side limit is not equal to the right hand side limit. Hence it is jump discontinuity. That is

\[\mathop {\lim }\limits_{x \to {1^ - }} f(x) \ne \mathop {\lim }\limits_{x \to {1^ + }} f(x)\]

If both the one sided limits are equal and it is not equal to f(x) at x=a then it is a removable discontinuity. That is

\[\mathop {\lim }\limits_{x \to {a^ - }} f(x) = \mathop {\lim }\limits_{x \to {a^ + }} f(x)\] and

\[\mathop {\lim }\limits_{x \to a} f(x) \ne f(a)\].

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Give a reason for the establishment of the Mohammedan class 10 social science CBSE

What are the two main features of Himadri class 11 social science CBSE

The continent which does not touch the Mediterranean class 7 social science CBSE

India has form of democracy a Direct b Indirect c Presidential class 12 sst CBSE

which foreign country is closest to andaman islands class 10 social science CBSE

One cusec is equal to how many liters class 8 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Which foreign country is closest to Andaman Islands class 11 social science CBSE

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