Continuous partial derivatives

I suspect that a better question is "What are the conditions that guarantee that the partials are continuous?"In summary, the conversation is about the conditions for partial derivatives to be continuous, with an example and a request for higher order partials. The question is clarified as asking for the conditions that guarantee the continuity of the partials.
  • #1
omri3012
62
0
Hallo,
What is the condition for partial derivatives to be continuous (if I have function f(x,y))?
Thanks,
Omri
 
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  • #2
Is the partial derivative a continuous function?

For example, does [itex]{\partial y}/{\partial x} = f(x,y)[/itex]?

If so, then what is the condition that f(x,y) be continous?

and then apply this to higher order partials [itex] {\partial^n y}/{\partial x^n} [/itex]
 
  • #3
omri3012 said:
Hallo,
What is the condition for partial derivatives to be continuous (if I have function f(x,y))?
Thanks,
Omri
That question is a bit strange. There are a number of things that are true if the partials are continuous (mixed derivatives the same, for instance) but I don't know of any that says "IF a condition is true THEN the partials are continuous".
 

1. What is the definition of a continuous partial derivative?

A continuous partial derivative is a mathematical concept used in multivariable calculus to describe the rate of change of a function in multiple variables. It is defined as the derivative of a function with respect to one variable, while holding all other variables constant. In other words, it measures how much a function changes as one variable changes, while keeping all other variables fixed.

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

A regular derivative is the rate of change of a function with respect to one variable. However, in multivariable calculus, there are multiple variables that can affect a function, so the concept of a partial derivative is used. A continuous partial derivative takes into account the changes of a function with respect to one variable, while keeping all other variables constant.

3. Can a function have a continuous partial derivative at a point but not be continuous at that point?

Yes, this is possible. A function can have a continuous partial derivative at a point, meaning the partial derivative exists and is defined at that point. However, the function may not be continuous at that point if the limit of the function does not exist or is not equal to the value of the function at that point.

4. How do you calculate a continuous partial derivative?

To calculate a continuous partial derivative, you use the same rules and formulas as you would for a regular derivative. However, when taking the derivative with respect to a specific variable, you treat all other variables as constants. This means you differentiate the function with respect to that variable while treating the other variables as fixed values.

5. Why are continuous partial derivatives important in science and mathematics?

Continuous partial derivatives are important because they allow us to analyze how a function changes in multiple variables. This is crucial in fields such as physics, economics, and statistics, where variables are often interdependent. Continuous partial derivatives also help us find maximum and minimum values of a function, which is useful in optimization problems.

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