# Continuity and Differentiability

#### Continuity

• A function ‘f’ is said to be continuous at x = a iff ‘f’ is defined in some neighborhood of ‘a’ and limit of f(x) when x tends to ‘a’ is equal to f(a).

• Let f be real valued continuous function ....
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#### Differentiability

- A function f(x) is differentiable at a point x = c, if its left hand derivative as well as right hand derivative exists finitely and are equal.
- The derivative of the function y = f(x) at any point x = c represents the slope of the tangent to the curve. So, slope of the tangent .... Read More

#### Differentiation of Composite and Implicit Functions

- The function whose values are found from two given functions by applying one function to an independent variable and then applying the second function to the result.
- If we have two functions u(x) and v(x), we can define a composite function
- w(x) = (u o v) (x) = u(v(x .... Read More

#### Differentiation of Inverse Trigonometric and Parametric Functions

- The derivatives of the inverse trigonometric functions are actually algebraic functions.
- Inverse trigonometric functions a .... Read More

#### Differentiation of Logarithmic and Exponential Functions

- The function y = f(x) = ax is called the exponential function with positive base, or a >1.
- The domain of an exponential function is the set of all real numbers.
- The range of the exponential function is the set of all positive real numbers.
- The derivativ .... Read More

#### Indeterminate Forms

If the limits of f(x) and g(x) are equal to 0, as x tends to a, then f(x)/ g(x) is said to assume indeterminate form 0/0.

Some other indeterminate forms are ∞/∞, ∞ - ∞, 0×∞, 00, 1∞ and ∞0.

**L’ Höpital’s rule< ....
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#### Mean-Value Theorem

- Rolle’s Theorem: Let a function be f: [a, b]®R, such that

- f is continuous over the interval [a, b],
- f is differentiable over the interval (a, b) and
- f(a) = f(b).

- Then there exists at least one real number, .... Read More

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