Class 12 Maths Revision Notes for Continuity and Differentiability of Chapter 5

CBSE Class 12 Mathematics Chapter 5 Continuity and Differentiability

• Continuity of function at a point: Geometrically we say that a function  is continuous at  if the graph of the function  is continuous (without any break) at .
• A funciton  is said to be continuous at a point  if:

(i)      exists i.e.,  is finite, definite and real.
(ii)     exists.
(iii)   

• A function  is continuous at   if  where  through positive values.
• Continuity of a function in a closed interval: A function  is said to be continuous in the closed interval  if it is continuous for every value of  lying between a and b continuous to the right of a and to the left of  i.e.,  and 
• Continuity of a function in a open interval: A function  is said to be continuous in an open interval  if it is continuous at every point in  .
• Discontinuity (Discontinuous function): A function  is said to be discontinuous in an interval if it is discontinuous even at a single point of the interval.
• Suppose  is a real function and  is a point in its domain. The derivative of  at  is defined by  provided this limit exists.
• A real valued function is continuous at a point in its domain if the limit of the function at that point equals the value of the function at that point. A function is continuous if it is continuous on the whole of its domain.
•  is derivative of first order and is also denoted by  or .
• Sum, difference, product and quotient of continuous functions are continuous. i.e., if f and g are continuous functions, then  is continuous. (f . g) (x) = f (x) . g(x) is continuous.

 (wherever g (x) ≠ 0) is continuous.

• Every differentiable function is continuous, but the converse is not true.
• Chain rule is rule to differentiate composites of functions. If f = v o u, t = u (x) and if both exist then
• Following are some of the standard derivatives (in appropriate domains):
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•  [Product Rule]
• , wherever  [Quotient Rule]
• If  then  [Chain Rule]
• If , then  [Parametric Form]
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• Logarithmic differentiation is a powerful technique to differentiate functions of the form    Here both f (x) and u (x) need to be positive for this technique to make sense.
• If we have to differentiate logarithmic funcitons, other than base , then we use the result  and then differentiate R.H.S.
• While differentiating inverse trigonometric functions, first represent it in simplest form by using suitable substitution and then differentiate simplified form.
• If we are given implicit functions then differentiate directly w.r.t. suitable variable involved and get the derivative by readusting the terms.
•  is derivative of second order and is denoted by  or .
• Rolle’s Theorem: If f: [a, b] → R is continuous on [a, b] and differentiable on (a, b) such that f (a) = f (b), then there exists some c in (a, b) such that f ′(c) = 0.
• Lagrange’s Mean Value Theorem: If f: [a, b] → R is continuous on [a, b] and differentiable on (a, b). Then there exists some c in (a, b) such that