CBSE Class 12 Maths Formula for Chapter-5 Continuity and Differentiability

Let us see the conditions where the function is said to be continuous,

• The real function ‘f’ is said to be left continuous at x = c, if there exists a function f(c), \[\lim_{x \rightarrow c^{-}} f(x)\] exists, and \[\lim_{x \rightarrow c^{-}} f(x) = f(c)\].

• The real function ‘f’ is said to be left continuous at x = c, if there exists a function f(c),\[\lim_{x \rightarrow c^{+}} f(x)\] exists, and \[\lim_{x \rightarrow c^{+}} f(x) = f(c)\].

• The real function ‘f’ is said to be continuous at x = c, if there exists a function f(c), \[\lim_{x \rightarrow c} f(x)\] exists, and \[\lim_{x \rightarrow c} f(x) = f(c)\].

In an open interval (a,b), a function ‘f’ is said to be continuous only if the function ‘f’ is continuous at every point in between that particular interval. In the closed interval [a,b] a function ‘f’ is said to be continuous along with that interval. If a function is continuous at a set of all the points then it is known as the domain of continuity. The domain of continuity is the proper subset of the domain of a function.

Properties and the Formula of Continuity and Differentiability Class 12:

• Property 1: If the two functions f, g are said to be continuous at x = c, only if it satisfies all the conditions.

1. α f is continuous at x = c, for all R.

2. f + g is continuous at x = c.

3. f - g is continuous at x = c.

4. fg is continuous at x = c.

5. f/g is continuous at x = c, where g(c) ≠ 0.

• Property 2: D₁ and D₂ are said to be the domains of continuity of functions ‘f’ and ‘g’ respectively, it should satisfy the following functions,

1. α f is continuous on D₁, for all α ∈ R.

2. f + g is continuous at D₁ ⋂ D₂.

3. f - g is continuous at D₁ ⋂ D₂.

4. fg is continuous at D₁ ⋂ D₂.

5. f/g is continuous at D₁ ⋂ D₂, where g(c) ≠ 0.

• Property 3: A polynomial function is said to be continuous everywhere.

• Property 4: A rational function is said to be continuous in every point in its particular domain.

• Property 5: If ‘f’ is said to be continuous at c, then |f| is continuous at x = c.

• Property 6: If ‘f’ is said to be continuous one-one function that is defined in the closed interval [a, b] in a range [c, d] then, f^-1:[c, d] ➝ [a, b].

• Property 7: If ‘f’ is said to be continuous at c and ‘g’ is continuous at f(c) then gof is said to be continuous at c.

• Property 8: All the basic trigonometric functions are continuous.

• Property 9: All the inverse trigonometric functions are continuous

• Property 10: Theorem, “If a function is continuous at any particular point, then it is necessarily continuous at that point”.

Till now we have studied about the properties and definitions of continuity and differentiability with formulas. Now let us learn the types of discontinuity, there are two main types of discontinuity:

1. Removable discontinuity.

2. Non-removable discontinuity.

All Formulas of Continuity and Differentiability Class 12

Derivative of a function ‘f’ is a real function and ‘c’ is a point in that domain then f at c is [f(c + h) - f(c)]/h it can be represented as f¹(c) or  [d f(x)]/dx.

Some of the rules has to be followed to find the differentiation of a function:

1. Algebra of Derivatives:

• (u ± v)¹ = u¹ + v¹

• (uv)¹ = u¹v + v¹u It is known as product rule.

• (u/v)¹ = [(u¹v) - (v¹u)]/v² It is known as quotient rule.

2. Chain Rule:

\[\frac{df}{dx} = \frac{dv}{du} \frac{du}{dx}\]

Along with the rules methods has to be followed, methods of continuity and differentiability formulas, are as follows,

1. Parametric form: To differentiate y = g(t) and x = f(t) separately by ‘t’, dy/dx = (dy/dt) (dt/dx), using this result we can find dy/dx.

2. Function of the Form g(x)^f(x): If the function is in the form of y = g(x)^f(x) and the f(x) and the g(x) is a differentiable function then log can be applied to obtain the form,

Log y = f(x) log g(x), this can be differentiated using product and chain rule.

3. Implicit Function: If ‘y’ cannot be expressed as in terms of x, i.e as y = f(x) then this function is said to be an implicit function.

4. Inverse trigonometric function.

5. Derivative of Order Two and Three: The second derivative is the derivative form of first, it can be represented by f¹, f². The derivative of the second derivative is known as the third derivative.

6. Mean Value Theorem: If f:[a, b] is continuous in the interval [a, b] and can be differentiable in (a, b) then f(a) = f(b), then f¹(c) = 0.

7. Rolle’s Theorem: If f:[a,b] is continuous in the interval [a, b] and can be differentiable in (a,b) then f(a) = f(b), then f¹(c) = [f(b) - f(a)]/(b - a).

Continuity and Differentiability Class 12 Formulas

Conclusion:

Some of the important concepts covered in this chapter include differentiability, continuity, logarithmic functions, exponential functions, and mean value theorem. It is one of the important chapters in class 12 and it can be continued in higher classes also. For more information on class 12 maths chapter 5 all formulas visit the vedantu official website, and refer to the notes available in the website.