Continuity
One Shot
Nishant Vora
B.Tech - IIT Patna
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Continuity
at a point
Layman Definition
Any function f(x) is said to be continuous if we can draw the graph of
that function without lifting pen
Mathematical Definition
Continuity at a point
A function f(x) is said to be continuous at x = a, if
Continuity at a point
4
3
5
5
4
4
3
3
Continuity at a point
4
3
3
Continuity at a point
If the function f(x) =
(JEE Main 2021)
is continuous at x = 0, then
A.
-5
B.
5
C.
-4
D.
4
is equal to :
Let a, b ∈ R, b ≠ 0, Define a function
(JEE Main 2021)
If f is continuous at x = 0, then 10 - ab is equal to.
Let a function f : R ⟶ R be defined as
(JEE Main 2021)
Where [x] is the greatest integer less
than or equal to x. If f is continuous on
R, then (a + b) is equal to :
A.
4
B.
3
C.
2
D.
5
Let f : R ⟶ R be defined as
(JEE Main 2021)
If f is continuous at x = 0, then α is equal to :
A.
1
B.
3
C.
0
D.
2
Let f : R ⟶ R be defined as
(JEE Main 2021)
where [x] is the greatest integer less than or equal to x. If f is
continuous at x = 2, then λ + μ is equal to
A.
e (-e + 1)
B.
e (e - 2)
C.
e
D.
2e - 1
(JEE Main 2021)
If f is continuous at x = 0, then the value of 6a + b2 is equal to :
A.
1-e
B.
e-1
C.
1+e
D.
e
Let f : R → R be defined as f(x) =
If f(x) is continuous on R, then a+b equals :
A.
3
B.
-1
C.
-3
D.
1
(JEE Main 2021)
(JEE Main 2019)
A.
2/π+5
B.
-2 / π + 5
C.
2/π-5
D.
2/5-π
If the function f defined on (-1/3, 1/3) by
(JEE Main 2020)
is continuous, then k is equal to _.
If the function f(x) =
is continuous at each point in its
domain and f(0) = 1/k, then k is _
(JEE Main 2021)
(JEE Main 2019)
A.
B.
C.
D.
3 Important Points
to Remember
1. Continuity is always talked in the domain of
function
⇒
⇒
f(x)=
are all continuous functions in its domain
is discontinuous at x = 1,
is discontinuous at x = 0.
2. Point function are continuous.
E.g.
3. Inverse of a discontinuous function
cannot be continuous.
3 Reasons for
Discontinuity
Reasons of Discontinuity:
1.
does not exist
Examples
1. f(x) = [x]
2. f(x) =
Reasons of Discontinuity:
2.
exist but is not equal to f(a)
Reasons of Discontinuity:
3.
f(a) is not defined
f(x) =
Reasons of Discontinuity:
Reasons of Discontinuity:
Note :
1. Every polynomial function is continuous.
2. Every rational function is continuous at every point where its
denominator is not equal to zero.
3. Logarithmic functions, exponential functions, trigonometric
functions, inverse circular functions, and modulus functions are
continuous in their domain.
Types of
Discontinuity
Identify the type of discontinuity ?
at x = 1
Identify the type of discontinuity ?
Identify the type of discontinuity ?
Identify the type of discontinuity ?
f(x) = [x] + [-x]
Identify the type of discontinuity ?
Identify the type of discontinuity ?
Identify the type of discontinuity ?
Identify the type of discontinuity ?
i.
at x = 1
Identify the type of discontinuity ?
ii.
f(x) = 2tanx at x =
Identify the type of discontinuity ?
iii.
at x = 0
Identify the type of discontinuity ?
at x = 0
Continuity in an
Interval
Continuity in open interval :
A function f is said to be continuous in (a, b) if f is continuous at
each & every point in (a, b)
Continuity in close interval :
A function f is said to be continuous in a closed interval [a, b] if
1. f is continuous at each and every point in (a, b)
2. f is right continuous at ‘a’ i.e.
3. f is left continuous at ‘b’ i.e.
f(x) = f(a) = finite quantity
f(x) = f(b) = finite quantity
Continuity in close interval :
How to calculate continuity in an interval ?
Look for possible points of discontinuity (PPOD)
Points to remember
All Polynomials, Trigonometric functions, Exponential and
Logarithmic functions are continuous in their domains.
#Bharose_wale_functions
Points to remember
Continuity of {f(x)} and [f(x)] should be checked at all points
where f(x) becomes integer.
Points to remember
Continuity of sgn (f(x)) should be checked at the points where f(x) = 0
Discuss the continuity of
f(x) = sgn(x3 - x)
f(x) = sgn(2cos x - 1)
f(x) = sgn(x2 - 2x + 3)
Points to remember
Continuity of a function should be checked at the points where
definition of a function changes
Points to remember
Continuity of a function should be checked at end Points
Let [t] denote the greatest integer ≤ t. The number of points where
the function
is not continuous is.
(JEE Main 2021)
If f : R ⟶ R be a function defined by
where [.] denotes the greatest integer function, then f is :
(JEE Main 2021)
A.
discontinuous only at x = 1
B.
discontinuous at all integral values of x except at x = 1
C.
continuous only at x = 1
D.
continuous for every real x
Let
, for - 10 < x < 10, where [t] denotes the greatest
integer function. Then the number of points of discontinuity of f is
equal to _.
(JEE Main 2020)
Let [t] denote the greatest integer ≤ t and
Then the function, f(x) = [x2] sin(πx) is discontinuous, when x is equal to :
(JEE Main 2020)
(JEE Main 2019)
Where [t] denotes the greatest integer less than or equal to t. Then, f is
discontinuous at :
A.
only one point
B.
only two points
C.
only three points
D.
four or more points
Theorems on
Continuity
Theorem 1: Super Table🔥
f(x)
g(x)
f(x) ± g(x)
f(x).g(x) or f(x)/g(x)
Continuous
Continuous
Continuous
Continuous
Continuous
Discontinuous
Discontinuous
May be Continuous
Discontinuous
Discontinuous
May be Continuous
May be Continuous
Theorem 2: Intermediate Value Theorem (IVT)
If f is continuous on [a, b] and f(a) ≠ f(b)
then for any value c ∈ (f(a), f(b)), there is at least one number x0 in (a, b)
for which f(x0) = c
Application of Intermediate Theorem
If f(x) is continuous in [a, b] and f(a).f(b) < 0
then there exist at least one root of f(x) in (a, b)
Theorem 3 : Extreme Value Theorem
If f is continuous on [a, b]
then f takes on, a least value of m and a greatest value M
on this interval
Key Points
1
If a function f is continuous on a closed interval [a, b] then it is
bounded.
Key Points
2
A continuous function whose domain is some closed interval
must have its range also in closed interval
Key Points
3
If f is continuous and onto and one-one on [a, b] then f-1(from
the range of f) is also continuous.
Key Points
4
If f(a) and f(b) possess opposite signs then ヨ at least one root of
the equation f(x) = 0 in the open interval (a, b) provided f is
continuous in [a, b]
Show that there exist at least one root f(x) in (0,π) f(x) = cos(x) - x2
Identify the intervals in which 2x3 - 6x + 1 = 0 has at least one root
A.
(0, 1)
B.
(1, 2)
C.
(-1, 0)
D.
(-2, -1)
Multi-correct ✅
Continuity of function involving
Today’s Topics
Continuity of functions involving limit
Remember,
Discuss the continuity of the following function
Find the value of x where f(x) =
is discontinuous
Discuss the continuity of f(x) in [0, 2], where f(x) =
Shortcut
Continuity of functions in which f(x) is defined differently for
rational and irrational value of x.
Discuss the continuity of the following function
Find the value of x where f(x) =
is discontinuous
🔥 Shortcut
Find the points where f(x) is continuous
i.
f(x) =
ii.
f(x) =
iii.
f(x) =
Continuity of
Composite Functions
Continuity of composite functions
if f (x) =
and g(x) =
then discuss the continuity of f(x), g(x) and fog(x)
Continuity of composite functions
if f (x) =
and g(x) =
then discuss the continuity of f(x), g(x) and fog(x)
gof(x)
Continuity of composite functions
if f (x) =
and g(x) =
then discuss the continuity of f(x), g(x) and fog(x)
fog(x)
Let the functions f : R ⟶ R and g : R ⟶ R be defined as :
(JEE Main 2021)
Then, the number of points in R where (f o g) (x) is not
CONTINUOUS is equal to :
A.
3
B.
1
C.
0
D.
2
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