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Directional Continuity and Continuity over an Interval is considered one of the most asked concept.
53 Questions around this concept.
The function is discontinuous only at such that - .The total number of such functions is:
is continuous from right at the point , then equals
The number of points of discontinuity of (where [ ] denotes the greatest integer function and is fractional part of ) in the interval ,is
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The function in does not take the value
Let be a continuous function defined for . If takes rational values for all and , then the value of is:
Let f is a continuous function in [a, b], g is a continuous function in [b, c]. A function h(x) is defined as
,( where the greatest integer less than or equal to ). Then the number of proints, where is discontinuous is
Let and
where denotes signum function of . If is discontinuous at exactly one point, then which of the following is not possible?
A discontinuous function satisfying is given by ________ for .
Let , where [x] is the greatest integer less than or equal to x. Then the number of points in (1,2) where f is discontinuous is:
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
A function y = f(x) is right - continuous at x = a if
For example,
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
Over a closed interval [a, b]
A function f(x) is continuous over a closed interval of the form [a, b] if
i.e.At x = a, we need to check . L.H.L. should not be evaluated to check continuity x = a
And at x = b, we need to check . 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 : (as in close left neighbourhood of x = c, the function equals 2)
RHL at x = c: (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
So f(x) is left continuous at x = 2
Condition 3
Left continuity at x = 3
f(3) = 3 and
(as in left neghbourhood 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
A function that is non-continuous, 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 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. .
Again removable discontinuity can be classified into missing point discontinuity and isolated point discontinuity
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
It makes the function continuous as now LHL = RHL = f(2)
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