## CBSE Class 12 Maths Notes Chapter 5 Continuity and Differentiability

**Continuity at a Point:** A function f(x) is said to be continuous at a point x = a, if

Left hand limit of f(x) at(x = a) = Right hand limit of f(x) at (x = a) = Value of f(x) at (x = a)

i.e. if at x = a, LHL = RHL = f(a)

where, LHL = \(\lim _{ x\rightarrow { a }^{ – } }{ f(x) }\) and RHL = \(\lim _{ x\rightarrow { a }^{ + } }{ f(x) }\)

Note: To evaluate LHL of a function f(x) at (x = o), put x = a – h and to find RHL, put x = a + h.

**Continuity in an Interval:** A function y = f(x) is said to be continuous in an interval (a, b), where a < b if and only if f(x) is continuous at every point in that interval.

- Every identity function is continuous.
- Every constant function is continuous.
- Every polynomial function is continuous.
- Every rational function is continuous.
- All trigonometric functions are continuous in their domain.

**Standard Results of Limits**

**Algebra of Continuous Functions**

Suppose f and g are two real functions, continuous at real number c. Then,

- f + g is continuous at x = c.
- f – g is continuous at x = c.
- f.g is continuous at x = c.
- cf is continuous, where c is any constant.
- (\(\frac { f }{ g }\)) is continuous at x = c, [provide g(c) ≠ 0]

Suppose f and g are two real valued functions such that (fog) is defined at c. If g is continuous at c and f is continuous at g (c), then (fog) is continuous at c.

If f is continuous, then |f| is also continuous.

**Differentiability:** A function f(x) is said to be differentiable at a point x = a, if

Left hand derivative at (x = a) = Right hand derivative at (x = a)

i.e. LHD at (x = a) = RHD (at x = a), where Right hand derivative, where

Note: Every differentiable function is continuous but every continuous function is not differentiable.

**Differentiation:** The process of finding a derivative of a function is called differentiation.

**Rules of Differentiation**

**Sum and Difference Rule:** Let y = f(x) ± g(x).Then, by using sum and difference rule, it’s derivative is written as

**Product Rule:** Let y = f(x) g(x). Then, by using product rule, it’s derivative is written as

**Quotient Rule:** Let y = \(\frac { f(x) }{ g(x) }\); g(x) ≠ 0, then by using quotient rule, it’s derivative is written as

**Chain Rule:** Let y = f(u) and u = f(x), then by using chain rule, we may write

**Logarithmic Differentiation:** Let y = [f(x)]^{g(x)} ..(i)

So by taking log (to base e) we can write Eq. (i) as log y = g(x) log f(x). Then, by using chain rule

**Differentiation of Functions in Parametric Form:** A relation expressed between two variables x and y in the form x = f(t), y = g(t) is said to be parametric form with t as a parameter, when

(whenever \(\frac { dx }{ dt } \neq 0\))

Note: dy/dx is expressed in terms of parameter only without directly involving the main variables x and y.

**Second order Derivative:** It is the derivative of the first order derivative.

**Some Standard Derivatives**

**Rolle’s Theorem:** Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b) such that f(a) = f(b), where a and b are some real numbers. Then, there exists at least one number c in (a, b) such that f'(c) = 0.

**Mean Value Theorem:** Let f : [a, b] → R be continuous function on [a, b]and differentiable on (a, b). Then, there exists at least one number c in (a, b) such that

Note: Mean value theorem is an expansion of Rolle’s theorem.

Some Useful Substitutions for Finding Derivatives Expression