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

## Class 12 Maths Revision Notes for Continuity and Differentiability of Chapter 5 â€“ Free PDF Download

Free PDF download of Class 12 Maths revision notes & short key-notes for Continuity and Differentiability of Chapter 5 to score high marks in exams, prepared by expert mathematics teachers from latest edition of CBSE books.

## CBSE Class 12 Mathematics Chapter 5 Continuity and Differentiability

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

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

• A functionÂ  Â is continuous atÂ  Â  ifÂ  Â whereÂ  Â through positive values.
• Continuity of a function in a closed interval: A functionÂ  Â is said to be continuous in the closed interval Â if it is continuous forÂ every value ofÂ  Â lying between a and b continuous to the right of a and to the left ofÂ  Â i.e.,Â  Â andÂ
• Continuity of a function in a open interval: A functionÂ  Â is said to be continuous in an open intervalÂ  Â if it is continuous at every point inÂ Â  .
• Discontinuity (Discontinuous function): A functionÂ  Â is said to be discontinuous in an interval if it is discontinuous even at a single point of the interval.
• SupposeÂ  Â is a real function andÂ  Â is a point in its domain. The derivative ofÂ  Â atÂ  Â is defined byÂ  Â 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.
• Â is derivative of first order and is also denoted byÂ  Â orÂ  .
• Sum, difference, product and quotient of continuous functions are continuous. i.e., if f and g are continuous functions, thenÂ  Â is continuous. (f . g) (x) = f (x) . g(x) is continuous.

Â (wherever g (x) â‰  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 bothÂ  exist then
• Following are some of the standard derivatives (in appropriate domains):
• Â [Product Rule]
• , whereverÂ  Â [Quotient Rule]
• IfÂ  Â thenÂ  Â [Chain Rule]
• IfÂ  , thenÂ  Â [Parametric Form]
• Logarithmic differentiation is a powerful technique to differentiate functions of the formÂ  Â Â Â 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 baseÂ  , then we use the resultÂ  Â 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.
• Â is derivative of second order and is denoted byÂ  Â orÂ  .
• Rolleâ€™s Theorem:Â If f: [a, b] â†’ 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 â€²(c) = 0.
• Lagrangeâ€™s Mean Value Theorem: If f: [a, b] â†’ R is continuous on [a, b] and differentiable on (a, b). Then there exists some c in (a, b) such that