Continuity and Differentiability
Continuity and Differentiability Formulas
What is Continuity?

Continuity of a function states the characteristics of the function and its functional value. A function is said to be continuous if the curve has no missing points or breaking points in a given interval or domain, that is the curve is continuous at every point in its domain.

A function f(x) is known as a continuous function at a point x = a, in its domain if the following listed three conditions are satisfied
1. f (a) exists which means that the value of f (a) is finite.
2. Lim x→a f (x) exists, that is the righthand limit = lefthand limit, and both R.H.S and L.H.S are finite.

Lim x→a f (x) = f (a)

A function f(x) is said to be continuous in the given interval I that is equal to [x1, x2] only if the three conditions listed above are satisfied for each and every point in the given interval I.
Formal Definition of Continuity:
A function is said to be continuous in the closed interval [a, b] if
1. f is continuous in (a, b)
2. limx>a+ f(x) = f (a)
3. limx>b f(x) = f (b)
A function is said to be continuous in the open interval (a, b) if
f (x) is going to be continuous within the unbounded interval (a, b) if at any point within the given interval the function is continuous.
Geometrical Interpretation of Continuity :
Function f is going to be continuous at x = c if there’s no break within the graph of the function at the purpose ( c , f(c) ).
In an interval, a function is claimed to be continuous if there’s no break within the graph of the function within the entire interval.
Then when can a function be discontinuous?

A function f is discontinuous at x=a if any of the following is true:
1. limx>a+ f(x) and limx>a f(x) exist but are not equal.
2. limx>a+ f(x) and limx>a f(x) exists are both equal but not equal to f(a).
3. f (a) is not defined.
Therefore we arrive at a modified definition of continuity: A function f is continuous at x=a if: limx>a+ f(x) = limx>a f(x) = f(a)
Note:
Let f and g be two real functions and let c be a point in the common domain of f and g. If the functions f and g are both continuous at x=c then:
1. f+g is continuous at x=c.
2. fg is continuous at x=c.
3. f•g is continuous at x=c.
4. f/g is continuous at x=c given that g(c) is not zero.
For Better Understanding Let’s Go Through an Example!
What is Differentiability?

Function f(x) is said to be differentiable at the point x = an if the derivative of the function f ‘(a) exists at every point in its given domain.

Differentiability formula
The differentiability formula is defined by –
f’(a) = \[\frac{f(a+h)f(a)}{h}\]
If a function is continuous at a particular point then a function is said to be differentiable at any point x = a in its domain. The vice versa of this is not always true.
Here are the derivatives of the basic trigonometric functions (differentiability formulas)
Differentiability and Continuity Problems and solution
Here are a few Differentiability and Continuity Problems and solutions!
Question 1) List down the continuity and differentiability formulas.
Solution) The continuity and differentiability formulas are as follows
The differentiability problems can be solved using the formula
f’(a) = \[\frac{f(a+h)f(a)}{h}\]
For a function f to be continuous it should satisfy the three conditions given below
1. f (a) exists which means that the value of f (a) is finite.
2. Lim x→a f (x) exists, that is the righthand limit = lefthand limit, and both R.H.S and L.H.S are finite.
3. Lim x→a f (x) = f (a)
Question 2) Explain the continuity of the given function f(x).
Where , f(x ) = sin x . cos x
Solution)We know that cos x and sin x both are continuous functions. We also know that the product of any two continuous functions is also a continuous function.
Therefore, we can say that the function f(x) = sin x . cos x is also a continuous function.