Careers360 Logo
JEE Main Result 2025 Session 2 Paper 1 (Out) - Paper 2 Scorecard Soon at jeemain.nta.nic.in

Directional Continuity and Continuity over an Interval - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Directional Continuity and Continuity over an Interval is considered one of the most asked concept.

  • 67 Questions around this concept.

Solve by difficulty

Which of the following graphs shows that limit exits at x=1 ? 

Which of the following function is not continuous at all x being in the interval [1,3]?

Which is true for f(x)  =[x]  , where [ ]  stands for greatest integer function?

If f(x)={[x]1x1,x10x=1} where [x] denotes greater integer x, then f(x) is

Let f(x)=h(x)/g(x), where h and g are continuous function on open interval (a,b) which of the following statement is true?

Let ‘f’ be a continuous function on [1, 3] if ‘f’ takes only rational values for all x in [1,3] and f(2) = 20 then f(1) =  

 

At x=0,f(x)=1|x| has

UPES B.Tech Admissions 2025

Ranked #42 among Engineering colleges in India by NIRF | Highest Package 1.3 CR , 100% Placements | Last Date to Apply: 15th May

ICFAI University Hyderabad B.Tech Admissions 2025

Merit Scholarships | NAAC A+ Accredited | Top Recruiters : E&Y, CYENT, Nvidia, CISCO, Genpact, Amazon & many more

For |sgn(x)| find the correct statement:

The point where the graph of the function breaks is called the point of

JEE Main 2025 College Predictor
Know your college admission chances in NITs, IIITs and CFTIs, many States/ Institutes based on your JEE Main rank by using JEE Main 2025 College Predictor.
Use Now

In which of the following, x=1 is the only point of discontinuity?

Concepts Covered - 2

Directional Continuity and Continuity over an Interval

Directional Continuity and Continuity over an Interval

A function may happen to be continuous in only one direction, either from the "left" or from the "right".
A function y=f(x) is left continuous at x=a if

limxaf(x)=f(a) or limh0+f(ah)=f(a) or LHL=f(a)
A function y=f(x) is right continuous at x=a if

limxa+f(x)=f(a) or limh0+f(a+h)=f(a) or RHL=f(a)
For example,
f(x)=y=[x],( where, [.] is G.I.F ). is right continuous at x=2

f(2)=limx2+[x]=2
But it is NOT left continuous at x=2 as LHL=1 but f(2)=2. Hence f(2) does not equal LHL at x=2.

Continuity over an Interval

Over an open interval (a,b)
A function f(x) is continuous over an open interval (a,b) if f(x) is continuous at every point in the interval.

For any c(a,b),f(x) is continuous if

limxcf(x)=limxc+f(x)=f(c)
Over a closed interval [a, b]
A function f(x) is continuous over a closed interval of the form [a,b] if
it is continuous at every point in (a,b) and
is rightcontinuous at x=a and
is leftcontinuous at x=b.
i.e.At x=a, we need to check f(a)=limxa+f(x)(=limh0+f(a+h)= R.H.L. ). L.H.L. should not be evaluated to check continuity x=a

And at x=b, we need to check f(b)=limxbf(x)(=limh0+f(bh)= L.H.L. ). R.H.L. should not be evaluated to check continuity x=b

Consider one example,

f(x)=[x], prove that this function is not continuous in [2,3],
Sol.
Condition 1
For continuity in (2,3)
At any point x=c lying in (2,3),
f(c)=[c]=2( as c lies in (2,3) )
LHL at x=c:limxc[x]=2 (as in close left neighbourhood of x=c, the function equals 2 )
RHL at x=c : limxc+[x]=2 (as in close right neighbourhood of x=c, the function equals 2 )
So function is continuous for any c lying in (2,3). Hence the function is continuous in (2,3)
Condition 2
Right continuity at x=2
f(2)=2

limx2+f(x)=limx2+[x]=limh0+[2+h]=2
So f(x) is left continuous at x=2

Condition 3

Left continuity at x=3
f(3)=3 and

limx3[x]=limh0+[3h]=2

(as in the left neighbourhood of 3,f(x)=2 )
So f(3) does not equal LHL at x=3
hence f(x) is not left continuous at x=3

So third condition is not satisfies and hence f(x) is not continuous in [2,3]

Discontinuity and Removable Types Discontinuity

Discontinuity and Removable Types Discontinuity

A non-continuous function is said to be a discontinuous function.

There are various kinds of discontinuity at a point, which are classified as shown below:

1. Removable Discontinuity   

2. Non-Removable Discontinuity

Finite Type

Infinite Type

Oscillatory

Removable Discontinuity

In this type of discontinuity, the limit of the function limxaf(x) necessarily exists but it is either not equal to f(a) or f(a) is not defined. However, it is possible to redefine the function at x=a in such a way that the limit of the function at x=a is equal to f(a), i.e. limxaf(x)=f(a).

Again removable discontinuity can be classified into missing point discontinuity and isolated point discontinuity.

Consider the function f(x)=x24x2, where, x2

f(x)=(x+2)(x2)(x2)=(x+2),x2

In the above graph, observe that the graph has a hole (missing point) at x=2, which makes it discontinuous at x=2.

Here LHL=RHL=4
Here function f(x) is not defined at x=2, i.e. f(2) is not defined. However, we can redefine the function as

f(x)={x24x2,x24,x=2
It makes the function continuous as now LHL = RHL =f(2)

Study it with Videos

Directional Continuity and Continuity over an Interval
Discontinuity and Removable Types Discontinuity

"Stay in the loop. Receive exam news, study resources, and expert advice!"

Books

Reference Books

Directional Continuity and Continuity over an Interval

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

Page No. : 4.5

Line : 11

Discontinuity and Removable Types Discontinuity

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

Page No. : 4.3

Line : 4

E-books & Sample Papers

Get Answer to all your questions

Back to top