Continuity And Differentiability

 


Hence, is continuous at
Now,  

Hence. is.no] differetiable at

A function's continuity specifies its properties and the value it serves as a function.If a curve is continuous at every point inside its domain and does not have any missing points or breaking points, the function is said to be continuous in nature.

If all three of the given conditions are met, then a function f(x) is said to be a continuous function at the point x = an in its domain. 

The three conditions are mentioned below:

The value of f(a) must be finite

If Lim _x→an f (x) is existing, is it essential that the right hand limit and left hand limit should be equal to each other, and both of them should be finite

Then Lim _x→an f (x) = f (a)

Thus, a function f(x) is said to be continuous at all positions in the range [a, b], including the endpoints a and b.

Continuity of f at a is: Llim _x→a+ f(x) = f (a)

Continuity of f at b is: Lim _x→b- f(x) = f (b)

In case of differentiability, if the derivative of the function f'(a) exists at every point within its defined domain, then the function f(x) is said to be differentiable at the point x = a.

The formula for differentiability is: f(a) =[f(a+h)−f(a)h]/h

We have, ...(i) 
Taking log on both sides of (i), we get 
....(ii)
The point where the curve cuts the x-axis is (e, 0). 

Function continuity: This concept makes more sense when described in terms of limitations. If a real function f(x) is close to becoming f(c), then it may be said to be continuous at a position like x = c.

lim_{x -> a} f(x) = f(a)

Conditions for a function's continuity: Any function must fulfil the following requirements in order to be continuous:

• Only if f(a) is a real integer and the function f(x) is stated at x = a, is the function continuous.
• As x gets closer to a, the function has a limit.
• The function's limit as x gets closer to a must match the function's value at x = a.

A function f(x) is differentiable at a point x = a, if f ' (a), i.e., the derivative of the function exists at each point of its domain.

The differentiability of a function is represented as:

f ' (x) = f (x + h) – f(x) / h

If a function f is continuous at any point, the same function is also differentiable at any point x = c in its domain. However, vice versa is not always applicable.

Put we get




...(i) 
Differentiating (i) w.r.t. 'x', we get 

Finding a function's derivative, which defines the rate at which the output of the function varies in relation to its input, is the process of differentiation. 

Differentiation enables us to comprehend the rate of change of a function at any given time.

The limit of the ratio of a function's output change to input change as the input change gets closer to zero is known as the derivative of the function. 

In terms of geometry, the slope of the tangent line to the function at a particular location is represented by the derivative. 

We may determine this slope using the differentiation process, and we can then examine how the function behaves at that point.

Finding a function's derivative, which defines the rate at which the output of the function varies in relation to its input, is the process of differentiation. 

The pace at which a function changes in response to an independent variable is known as its derivative. They are characterised as the rate at which one quantity y changes in relation to another, x. 

The primary objects of differentiation are derivatives. 

Finding a function's derivative is the process of differentiation.

When a quantity is changing and the rate of change is variable, derivatives are utilised.

It is employed to determine the dependent variable's sensitivity to the independent variable.

They have to do with the instantaneous pace at which one quantity changes in respect to another.