Continuity and Discontinuity is one of the important parts of Calculus, which applies to measuring the change in the function at a certain point. Mathematically, it forms a powerful tool by which graphs of functions are determined, the maximum and minimum of functions found, and problems on motion, growth, and decay, to name a few. These concepts of Continuity and Discontinuity have been broadly applied in branches of mathematics, physics, engineering, economics, and biology.
JEE Main: Study Materials | High Scoring Topics | Preparation Guide
JEE Main: Syllabus | Sample Papers | Mock Tests | PYQs
In this article, we will cover the concepts of Directional Continuity. This concept falls under the broader category of sets relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of two questions have been asked on this concept, including one in 2021, and two in 2023.
A real function
This definition requires a bit of elaboration. Suppose
Observe that
When a graph breaks at a particular point when it approaches from left and right.
So limit exists but is not continuous: but when it is equal to
1.
2. Right hand limit at
3. Left hand limit at
So continuity can be defined in two ways: Continuity at a point and Continuity over an interval.
A function
A function is said to be directional continuous at a point when the limit of the function exists and is continuous at all or specified directions.
Over an open interval
A function
For any
Over a closed interval
A function
- it is continuous at every point in
- is right-continuous at
- is left-continuous at
i.e.At
L.H.L. should not be evaluated to check continuity of the first element of the interval,
Similarly, at
R.H.L. should not be evaluated to check continuity of the last element of the interval
Consider one example,
Sol.
Condition 1
For continuity in
At any point
LHL at
RHL at
So function is continuous for any c lying in
Condition 2
Right continuity at
So
Condition 3
Left continuity at
(as in left neghbourhood of
So
hence
So the third condition is not satisfied and hence
Example 1:
1)
2)
3)
4)
Solution:
As we have learned
Continuity in an open interval -
Hence, the answer is the option 1.
Example 2:
1)
2)
3)
4) All of them
Solution:
As we have learned
Continuity in an open interval -
At every
Hence, the answer is the option 1
Example 3: Which of the following statements is false?
1)
2)
3)
4)
Solution: To check left continuity we need to find LHL and function value at the point
(A)
Hence, the answer is the option 2.
Example 4: Which of the following statements is false? ([.]= G.I.F)
1)
2)
3)
4)
Solution:
Continuity from Right -
The function
so
but in
Hence, the answer is the option 3.
Example 5 : Which of the following functions is not continuous at all
1)
2)
3)
4)
Solution:
As we have learned
Continuity from Right -
1.
2. Right hand limit at
3. Left hand limit at
(A),(B),(C) are the functions which are continuous at every point in
so (A),(B),(C ) are continuous at every point of
In (D),
The condition for the discontinuity:
i)
ii)
Conditions for the continuity are:
i)
ii) Right hand limit at
iii) Left hand limit at
The function
The function
14 Feb'25 09:37 PM
14 Feb'25 09:18 PM
14 Feb'25 09:15 PM
14 Feb'25 09:11 PM
14 Feb'25 09:05 PM
14 Feb'25 09:02 PM
14 Feb'25 09:00 PM
14 Feb'25 08:57 PM
14 Feb'25 08:54 PM
14 Feb'25 08:23 PM