Differentiability of Special Functions

Lets consider some special functions are:

1. f(x) = [x], which is the greatest integer of x, and the other one
2. f(x) = {x}, which is the fractional part of x

For f(x) = [x]

So, first, we go with f(x) = [x], to check the differentiability of the function we have to plot the graph first. So let’s plot the graph

So as we see in the graph that between 0 and 1 the value of the function is 0 and between 1 and 2 the value of the function is 1 and between 2 and 3 the value of the function is 2, Similarly on the -ve side between -1 and 0 the value of the function is -1. So if we talk about the domain of the function the domain is the entire value of real values but the range of this function is only integers, the function will take only integer values because it is the greatest integer of x.

Now let us talk about the differentiability of this particular function. Firstly we are talking about integer points, so firstly we are checking the differentiability at integer points. As we know for the function to be differentiable the function should continuous first, as we see the graph at point x = 1,

At integer point,

Consider x =1 

RHL = lim_{x -> 1+} [x] = 1

LHL = lim_{x -> 1-} [x] = 0 

So, RHL ≠ LHL    

It is checked for x = 1, but it will valid for all the integer points result will be the same. [x] is not continuous at integer points, So it is not differentiable at integer points. So, [x] is not continuous at integer points. It is also not differentiable at integer points.

Now, let us find what is happening on non-integer points. Let us consider x = 2.5, let’s find RHL and LHL

RHL = lim_{x -> 2.5+} [x] = 2

LHL = lim_{x -> 2.5-} [x] = 2   

Since both RHL and LHL are equal, so our function [x] is continuous at a non-integer point. Now we have to check the differentiability at non-integer points, so we have to find the slope of the function which we can find by finding the derivative of the function [x] at point 2.5

f'(x) = d[x] / dx at x = 2.5 = 0 

Therefore, the function is differentiable at all non-integer points.

For f(x) = {x}

Now we are considering the second function which f(x) = {x} which is the fractional part of x. To find the differentiability and continuity we have to plot the graph first.

So in this graph, the domain of the function is the entire range of real values and the range of this function is only 0 to 1 because any fractional part of the value is between 0 to 1. Let’s find for integer values, Consider the point x = 1

RHL = lim_{x -> 1+} {x} = 0 

LHL = lim_{x -> 1-} {x} = 1   

Since RHL ≠ LHL function {x} is not continuous and hence not differentiable. Let us find for non-integer points. Considering x = 1.5

RHL = lim_{x -> 1.5+} {x} = 0.5 

LHL = lim_{x -> 1.5-} {x} = 0.5   

Since RHL = LHL, function is continuous. To find the differentiability we have to find the slope of the function which we can find by finding the derivative of the function [x] at point 2.5

f'(x) = d{x} / dx at x = 1.5 = 1   

Therefore, the function {x} is differentiable at non-integer points. 

Continuity or continuous which means, “a function is continuous at its domain if its graph is a curve without breaks or jumps”. A function is continuous at a point in its domain if its graph does not have breaks or jumps in the immediate neighborhood of the point.

Continuity at a Point:

A function f(x) is said to be continuous at a point x = a of its domain, if

(LHL) = (RHL) = f(a) or lim f(x) = f(a)

where 

• (LHL)_{x = a}  = lim_{x -> a-} f(x) and  
• (RHL)_{x = a} = lim_{x -> a+} f(x)

Note: To evaluate LHL and RHL of a function f(x) at x = a, put x – h and x + h respectively, where h -> 0

Discontinuity at a Point: 

If f(x) is not continuous at a point x = a, then it is discontinuous at x = a. 

There are various types of discontinuity:

• Removable discontinuity: If lim_{x -> a-} f(x) = lim_{x -> a+} f(x) ≠ f(a)
• Discontinuity of the first kind: If lim_{x -> a+} f(x) ≠ lim_{x -> a+} f(x)
• Discontinuity of the second kind: If lim_{x -> a-} f(x) or lim_{x -> a+} f(x) both do not exist