Class-12 . Continuity and differentiability Basic

 

Topic and subtopics covered in continuity and differentiability:

Sr. No.	Topic Name
1.	Continuity
2.	Differentiability
3.	Derivatives of Composite and Implicit Functions
4.	Logarithmic Differentiation
5.	Exponential and Logarithmic Functions
6.	Derivative of Functions in Parametric Forms
7.	Second Order Derivative
8.	Mean Value Theorem

Continuity.    function is said to be continuous if there is no break in the graph of the function in the entire interval.

( जब हम किसी function का ग्राफ बनाते हैं और वह ग्राफ बिना टूटे पूरा बन जाता हो ऐसे function को हम continuous  function कहते हैं )

                                        OR

A continuous function is a function for which small changes in the input results in small changes in the output. Otherwise, a function is said to be discontinuous.

A function f(x) is said to be continuous at x = a if

                                                       

                                      OR

$\begin{array}{l}\underset{�\to �}{lim}�(�)=\underset{�\to {�}^{+}}{lim}�(�)=�(�)\end{array}$

i.e., LHL = RHL = value of the function at x = a

Else, a function f(x) is said to be a discontinuous function.

Points to Remember

• Following functions are everywhere continuous:
  (a) A constant function
  (b) The identity function
  (c) A polynomial function
  (d) Modulus function
  (e) Exponential function
  (f) Sine and Cosine functions

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• Following functions are continuous in their domains:

• a) A logarithmic function
  (b) A rational function
  (c) Tangent, cotangent, secant and cosecant functions
  (d) All inverse trigonometric functions are continuous in their respective domains

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• A function f(x) is said to be continuous if it is continuous at every point on its domain.

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• Algebra of continuous functions:

  If the two real functions, say f and g, are continuous at a real number c, then

  (i) f + g is continuous at x=c.

  (ii) f – g is continuous at x=c.

  (iii) f. g is continuous at x=c.

  (iv)f/g is continuous at x=c, (provided g(c) ≠ 0).

A  function will be differentiable if:

The function f(x) is said to be differentiable at x = a, if

 Both RHD & LHD exist and are equal

Differentiation  means the rate of change of one quantity with respect to another.

                          OR

The ratio of a small change in one quantity with a small change in another which is dependent on the first quantity is called differentiation.

Differentiation Formulas
d/dx (a) = 0 where a is constant
d/dx (x) = 1
d/dx(xn) = nxn-1
d/dx sin x = cos x
d/dx cos x = -sin x
d/dx tan x = sec2 x
d/dx ln x = 1/x
d/dx ex = ex

• Rolle’s Theorem:
  Let f be a real value of function defined on the closed interval [a, b] such that
  (i) It is continuous on [a,b]
  (ii) It is differentiable on (a,b) and
  (iii) f(a)=f(b)
  Then, there exists at least one real number c∈(a,b) such that f'(c)=0.

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• Lagrange’s Mean Value Theorem:
  Let f(x) be a function defined on [a, b] such that
  (i) It is continuous on [a,b] and
  (ii) Differentiable on (a,b)
  Then, there exists at least one c∈(a,b) such that f'(c)=f(b)-f(a)/b-a