## Class 12 Maths Revision Notes for Continuity and Differentiability of Chapter 5 – 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 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):
- [Product Rule]
- , wherever [Quotient Rule]
- If then [Chain Rule]
- If , then [Parametric Form]
- 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