Continuity
01 Continuity of a Function at a Point
Suppose f is a real function on a subset of the real numbers & let c be a point in the
domain of f. Then f is continuous at c if
03
Differentiab
A function f is said to b
hand derivatives exist
Here at x = c,
Left Hand Derivative,
lim f(x)= f(c)
x c L.H.D. = lim f(c-h)-f(c
h 0
-h
Continuity of a Function in an Interval
Suppose f is a function defined on a closed interval [a, b], then for f to be continuous, it needs to be Right Hand Derivative,
continuous at every point in [a, b] including the end points a & b.
R.H.D. = lim f(c+h)-f(
h 0
h
Continuity of f at a, lim f(x)= f(a)
x a+
Theorem: If a function f is differentiable at
Continuity of f at b, lim f(x)= f(b) differentiable function is continuous, but th
x b-
A function which is not continuous at point x=c is said to be discontinuous at that point
Algebr
Algebra of Continuous Functions 04 Let u, v be the
02 Theorem 1: Suppose f & g be two real functions continuous
at a real number c, Then
(1) Sum and
(2) Leibnitz o
(3) Quotient
(1) f + g is continuous at x=c (3) f.g is continuous at x=c
(2) f - g is continuous at x=c (4) f/g is continuous at x=c,
(provided g(c)≠0)
Cha
05
Theorem 2: Suppose f & g are real valued functions such that (fog) is
If y is a f
defined at c. If g is continuous at c& if f is continuous at g(c), then
a functi
(fog) is continuous at c.
d
Then,
d
Implicit Functions
06 An equation of the form f(x, y) = 0 in which y
is not expressible in terms of x is called an
implicit function of x & y. CONTIN
DIFFER
Derivative of Implicit Functions
Let y=f(x, y), where f(x, y) be an implicit function of x & y.
Firstly differentiate both sides of equation w.r.t x
dy
Then take all terms involving on L.H.S. & remaining terms on
dx
R.H.S. to get the required value.
Logarithmic Differe
Differentiation of Inverse
08 Logarithmic Differentiation
differentiate functions of th
07 Trigonometric Functions f(x) & u(x) are positive.
f(x) f (x) Domain of f We apply logarithm (to base) on both sid
differentiate by using chain rule, in this wa
1
sin-1x (-1,1) is called logarithmic
1 - x2
((
d d 1 & d
ex = ex , (log x) =
-1 dx dx x dx
cos-1x (-1,1)
1 - x2
1
Derivatives of Func
09
tan-1x R
1 + x2
The set of equations x = f(t), y = g(t)
dy dy / dt g(t)
-1 Here, = or
cot x -1 R dx dx / dt f(t)
1 + x2
dy
Here, is expressed in terms of parameter only with
1 dx
sec-1x x >1
x x2 - 1
01 Continuity of a Function at a Point
Suppose f is a real function on a subset of the real numbers & let c be a point in the
domain of f. Then f is continuous at c if
03
Differentiab
A function f is said to b
hand derivatives exist
Here at x = c,
Left Hand Derivative,
lim f(x)= f(c)
x c L.H.D. = lim f(c-h)-f(c
h 0
-h
Continuity of a Function in an Interval
Suppose f is a function defined on a closed interval [a, b], then for f to be continuous, it needs to be Right Hand Derivative,
continuous at every point in [a, b] including the end points a & b.
R.H.D. = lim f(c+h)-f(
h 0
h
Continuity of f at a, lim f(x)= f(a)
x a+
Theorem: If a function f is differentiable at
Continuity of f at b, lim f(x)= f(b) differentiable function is continuous, but th
x b-
A function which is not continuous at point x=c is said to be discontinuous at that point
Algebr
Algebra of Continuous Functions 04 Let u, v be the
02 Theorem 1: Suppose f & g be two real functions continuous
at a real number c, Then
(1) Sum and
(2) Leibnitz o
(3) Quotient
(1) f + g is continuous at x=c (3) f.g is continuous at x=c
(2) f - g is continuous at x=c (4) f/g is continuous at x=c,
(provided g(c)≠0)
Cha
05
Theorem 2: Suppose f & g are real valued functions such that (fog) is
If y is a f
defined at c. If g is continuous at c& if f is continuous at g(c), then
a functi
(fog) is continuous at c.
d
Then,
d
Implicit Functions
06 An equation of the form f(x, y) = 0 in which y
is not expressible in terms of x is called an
implicit function of x & y. CONTIN
DIFFER
Derivative of Implicit Functions
Let y=f(x, y), where f(x, y) be an implicit function of x & y.
Firstly differentiate both sides of equation w.r.t x
dy
Then take all terms involving on L.H.S. & remaining terms on
dx
R.H.S. to get the required value.
Logarithmic Differe
Differentiation of Inverse
08 Logarithmic Differentiation
differentiate functions of th
07 Trigonometric Functions f(x) & u(x) are positive.
f(x) f (x) Domain of f We apply logarithm (to base) on both sid
differentiate by using chain rule, in this wa
1
sin-1x (-1,1) is called logarithmic
1 - x2
((
d d 1 & d
ex = ex , (log x) =
-1 dx dx x dx
cos-1x (-1,1)
1 - x2
1
Derivatives of Func
09
tan-1x R
1 + x2
The set of equations x = f(t), y = g(t)
dy dy / dt g(t)
-1 Here, = or
cot x -1 R dx dx / dt f(t)
1 + x2
dy
Here, is expressed in terms of parameter only with
1 dx
sec-1x x >1
x x2 - 1
