Continuity and Differentiability Class 12 Notes Mathematics Chapter 5

CHAPTER AT A GLANCE

1. Continuity of a function at a point – Let f be a real function and a be in the domain of f. We say f is continuous at a, if

2. Continuity of A function in an interval

3. Basic results on Continuous Functions

Note: The product of one continuous and one discontinuous function may or may not be continuous

4. Differentiability of a function at a point – f (x) is differentiable at (x = c), iff both Left and Right Hand Derivative at (x = c) exist and Left Hand Derivative at (x = c) = Right Hand Derivative at (x = c)

5. Differentiability of a function in a set

(i) Differentiability in an Open Interval A function f (x) defined in an open interval (a, b) is said to be differentiable or derivable in open interval (a, b) if it is differentiable at each point of (a, b).

(ii) Differentiability on a Closed Interval A function f (x) defined on [a, b]is said to be differentiable or derivable at the end points a and b of the interval (a, b) if it is differentiable from the right at a and from the left at b.

exists then f is differentiable at x = a from right and at x = b from left respectively. If f is derivable in the open interval (a, b ) and also at the end points a from right and at the end point b from left, then f is said to be derivable on the closed interval [a, b]

6. Theorems on Differentiability

(i) The addition of a differentiable and a non-differentiable functions is always non-differentiable.

(ii) The product of a differentiable and a non-differentiable functions maybe differentiable.

(iii) If both f(x) and g(x) are non-differentiable at x = a, then (f(x) + g(x)) maybe differentiable at x = a

7. Differentiability and Continuity

(i) If a function is differentiable at every point of its domain, then it must be continuous in that domain.

(ii) A continuous function may or may not be differentiable.

(iii) If a function is not continuous then it will not be differentiable.

8. The derivative of composite function (Chain Rule) – If y = f (z) and z = g (x) be two independent functions in which the independent variable of the first function is the dependent variable of the second function, then the derivative of y with respect to x is given by

9. The derivative of logarithmic function – If y = log a x, where x > 0 (a > 0 but a ¹ 1)

10. The derivative Of exponential function – If y = ax, where a > 0

11. The Derivative of A Function Represented Parametrically – Let x = f (t),y = g(t) be two derivable functions each defined in some interval. Here, t is the parameter. We also suppose that the function f is invertible and f denotes the inverse of f. Now, f being derivable its inverse, f is also derivable. We have

12. Differentiation of one Function w.r.t. other Function – Let u = f (x) andv = g (x) be two functions of x. Then to find the derivative of f (x) w.r.t. g(x),

13. The Derivative of an implicit function – If the relation between the variables x and y is given by an equation containing both, and this equation is not immediately solvable for y, then y is called an implicit function of x. Implicit functions are given by f (x, y) = 0. For example : x2y + sin (xy) + x cos y = 0

(i) In order to find dy/dx, in the case of implicit functions, we differentiate each term w.r.t. x regarding y as a functions of x and then collect terms in dy/dx together on one side to finally find dy/dx.

(ii) In answers of dy/dx in the case of implicit functions, both x and y represent.

14. Logarithmic Differentiation – If differentiation of an expression or an equation is done after taking log on both sides, then it is called logarithmic differentiation. This method is useful for

(i) The function whose base and power both are the functions of a variable like x i.e., the functions is of the form [f (x)] g (x)

(ii) The function which is either in the form of factors or quotient or both but so complex that directly using the product or quotient rule or both simultaneously is very lengthy

15. List of Differentiation of Some Standard Functions

16. Rolle’s Theorem – If a function f (x) is

  1. continuous in a closed interval [a, b]
  2. derivable (or differentiable) in the open interval (a, b) and
  3. f (a) = f (b), then there exists at least one real number c in (a, b) such that f ¢ (c) = 0

17. Langrange’s Mean Value Theorem – If a function f (x) is

  1. continuous in the closed interval [a, b] and
  2. derivable in the open interval (a, b),then there exists at least one real number c in (a, b) such that

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