Continuity and Differentiability

Continuity and Differentiability Formulas

Differentiability and Continuity is one of the most important topics and it helps students to understand  various concepts like continuity at a certain point, derivative of functions, and continuity on a given interval. (image will be updated soon)


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 right-hand limit = left-hand limit, and both R.H.S    and L.H.S are finite.

  1. 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.

In simpler words, if the function is undefined or does not exist, then we say that the function is discontinuous.


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. f-g 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!

We can show that the given function is continuous at x=4,

                        f(x) = (x² - 2x)/(x - 3)

Solution: The given function f(x) is continuous at x=4 because of the following –

  • f (4) exists – When we substitute the value of x = 4, we get the answer as 8.

  • f(x)   -  We can factorize the given function, 

f(x) = x(x-2)/(x-3)

Nothing is getting canceled, but when we substitute the value of x=4, then the value of the function results to 8.

  • f(4) =  f(x)

Since, both the sides R.H.S and L.H.S are equal to 8 when the value of x= 4, the function is continuous.

If any of the above conditions are not true, then the equation is said to be discontinuous.

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)- 

d/dx (sin x)     = 

cos x

d/dx (cos x)    =

- Sin x

d/dx (tan x)    =

Sec2x

d/dx (cot x)    =

cosec2x

d/dx (sec x)    =

Sec x tan x

d/dx (cosec x)    =

- cosec x cot x


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 right-hand limit = left-hand 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.