Continuous partial derivative?

In summary, the conversation discusses how some functions cannot be well approximated by tangent planes at a specific point, using the example of f(x)= xy/(x^2 + y^2) for x ≠ 0 and 0 for x = 0. The partial derivatives at (0,0) exist and are zero, but the function is not continuous at that point. The concept of a 'continuous partial derivative' in two variables is then brought up and the conversation concludes with a clarification that differentiability only implies the existence of partial derivatives, not the other way around.
  • #1
Shaybay92
124
0
My textbook describes how some functions are not well approximated by tangent planes at a particular point. For example

f(x)= xy / (x^2 + y^2) for x /= 0
0 for x = 0

at (0,0) the partial derivatives exist and are zero but they are not continuous at 0. What exactly is a 'continuous partial derivative' in two variables? How do you visualize this?
 
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  • #2
A partial derivative is a function, so speaking of continuity makes perfect sense.

This should help:

http://www.math.tamu.edu/~tvogel/gallery/node14.html
 
  • #3
In the case of that example, is it not differentiable at zero because its not continuous there?
 
  • #4
Shaybay92 said:
In the case of that example, is it not differentiable at zero because its not continuous there?

Correct.

A function is only differentiable at zero if a unique tangent plane can be assigned there.

Differentiability IMPLIES existence of partial derivatives, but the converse does not hold.
 
  • #5
Thanks! By the way, nice job on the 9,999 posts :)
 

1. What is a continuous partial derivative?

A continuous partial derivative is a mathematical concept that measures the rate of change of a function with respect to one of its variables, while holding all other variables constant. It is denoted by the symbol ∂ and is often used in multivariable calculus and optimization problems.

2. How is a continuous partial derivative different from a regular derivative?

A continuous partial derivative is similar to a regular derivative, but it only considers the change in one variable while keeping other variables constant. A regular derivative, on the other hand, considers the overall change in the function with respect to all variables.

3. Why is the concept of continuous partial derivative important in science?

The concept of continuous partial derivative is important in science because it allows us to analyze the behavior of complex systems and understand how different variables affect the overall function. It is used in fields such as physics, economics, and engineering to optimize systems and make predictions.

4. What are some real-life applications of continuous partial derivatives?

Continuous partial derivatives have many real-life applications, such as in optimization problems, where we need to find the maximum or minimum value of a function. They are also used in physics to calculate the rate of change of quantities such as velocity and acceleration, and in economics to analyze demand and supply functions.

5. How can I calculate a continuous partial derivative?

To calculate a continuous partial derivative, you need to first identify the variable you want to differentiate with respect to. Then, treat all other variables as constants and use the standard rules of differentiation to find the derivative. If you are unsure of the process, you can consult a calculus textbook or use online resources for step-by-step instructions.

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