# Demonstration of the differentiability of a continuous function

• Telemachus
In summary, to demonstrate that an average function is differentiable on all of its domain, you can use the limit with generic points (x_0,y_0) and show that the function is continuous and the limit exists and is finite.
Telemachus

## Homework Statement

I have some doubts about the demonstration of the differentiability. If I'm asked to proof that an average function is differentiable on all of it domain, let's suppose its a continuous function on all of its domain, but it has not continuous partial derivatives. How should I demonstrate that its differentiable? May I use the limit with generic points $$(x_0,y_0)$$? I mean, if I use this limit (the one with the function and the tangent plane over the square root that represents a disk), and its a differentiable function, with this generic points the limit should give zero, right?

Bye there, thanks for posting.

PD: I'm talking for function of two or more variables.

Homework EquationsThe definition of differentiability is that a function f is differentiable at a point (x_0,y_0) if the limit of lim_{(h,k)->(0,0)} (f(x_0 + h, y_0 + k) - f(x_0, y_0))/(h^2 + k^2)^1/2exists and is finite.The Attempt at a SolutionYes, you can use the limit with generic points (x_0,y_0) to demonstrate that an average function is differentiable on all of its domain. As long as the function is continuous on its domain and the limit exists and is finite, then the function is differentiable.

## What is the definition of differentiability for a continuous function?

Differentiability of a continuous function at a point means that the function has a well-defined derivative at that point. This means that the slope of the tangent line at that point exists and is unique.

## How is the differentiability of a continuous function determined?

The differentiability of a continuous function can be determined by taking the limit of the difference quotient as the interval approaches 0. If this limit exists and is finite, then the function is differentiable at that point.

## What is the relationship between differentiability and continuity for a function?

A continuous function is not necessarily differentiable, but a differentiable function is always continuous. This is because differentiability requires the function to be continuous, but also adds the condition of having a defined derivative.

## What is the importance of differentiability in calculus and real-world applications?

Differentiability is important in calculus as it allows us to estimate the behavior of a function near a particular point. It also plays a crucial role in optimization and finding maximum and minimum values of a function. In real-world applications, differentiability is used in physics, engineering, and economics to model and analyze various phenomena.

## Can a function be differentiable at one point but not at another?

Yes, a function can be differentiable at one point and not at another. This is because differentiability is defined at a specific point and does not necessarily extend to the entire function. A function may have a well-defined derivative at one point but not at another due to a discontinuity or sharp turn in the function.

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