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Directional Continuity and Continuity over an Interval is considered one of the most asked concept.
67 Questions around this concept.
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
Let
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
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For
The point where the graph of the function breaks is called the point of
In which of the following,
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
A function
For example,
But it is NOT left continuous at
Continuity over an Interval
Over an open interval
A function
For any
Over a closed interval [a, b]
A function
it is continuous at every point in
is rightcontinuous at
is leftcontinuous at
i.e.At
And at
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 the left neighbourhood of
So
hence
So third condition is not satisfies and hence
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
Again removable discontinuity can be classified into missing point discontinuity and isolated point discontinuity.
Consider the function
In the above graph, observe that the graph has a hole (missing point) at
Here
Here function
It makes the function continuous as now LHL
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