# Does differentiable imply continuous partial derivatives?

## Does differentiable imply continuous partial derivatives?

The differentiability theorem states that continuous partial derivatives are sufficient for a function to be differentiable. The converse of the differentiability theorem is not true. It is possible for a differentiable function to have discontinuous partial derivatives.

What does it mean to have continuous partial derivatives?

If a function has continuous partial derivatives on an open set U, then it is differentiable on U. A standard example is the function f(x)=x2sin(1x) which is differentiable but its partial derivative with respect to x f′(x)=2xsin(1x)−cos(1x) is not continuous.

### Is every continuous function differentiable give example?

We have the statement which is given to us in the question that: Every continuous function is differentiable. Therefore, the limits do not exist and thus the function is not differentiable. But we see that f(x)=|x| is continuous because limx→cf(x)=limx→c|x|=f(c) exists for all the possible values of c.

Why must function be continuous to be differentiable?

Yes differentiable function is always continuous because in a graph of function there is no sharp corner this means that it is going continuously.

## Are all differentiable functions continuous?

We see that if a function is differentiable at a point, then it must be continuous at that point. There are connections between continuity and differentiability. If is not continuous at , then is not differentiable at . Thus from the theorem above, we see that all differentiable functions on are continuous on .

What is continuous derivative?

The derivative of a function (if it exists) is just another function. Saying that a function is differentiable just means that the derivative exists, while saying that a function has a continuous derivative means that it is differentiable, and its derivative is a continuous function.

### How is the derivative of a differentiable function defined?

Consequently, the only way for the derivative to exist is if the function also exists (i.e., is continuous) on its domain. Thus, a differentiable function is also a continuous function. But just because a function is continuous doesn’t mean its derivative (i.e., slope of the line tangent) is defined everywhere in the domain. How so?

Can a continuous function be called a differentiable function?

It turns out that the answer to this question is: no. A differentiable function f for which f ′ is continuous is called continuously differentiable. This concept plays an important role in analysis, differential equations, and even vector calculus.

## What is the difference between continuity and differentiability?

Continuity and Differentiability. Differentiability is when we are able to find the slope of a function at a given point. In other words, a function is differentiable when the slope of the tangent line equals the limit of the function at a given point. What this really means is that in order for a function to be differentiable,…

How is x ( 1 / 3 ) differentiable at x = 0?

At x=0 the derivative is undefined, so x(1/3) is not differentiable. At x=0 the function is not defined so it makes no sense to ask if they are differentiable there. To be differentiable at a certain point, the function must first of all be defined there!

05/04/2021