When Are Partial Derivatives Continuous?

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SUMMARY

The continuity of partial derivatives is contingent upon the function being continuously differentiable. Specifically, for a function f(x,y) to have continuous partial derivatives, it must satisfy the condition of being differentiable in a neighborhood around the point of interest. This ensures that mixed partial derivatives are equal, as stated by Clairaut's theorem. The discussion highlights the lack of a definitive condition that guarantees continuity solely based on the existence of partial derivatives.

PREREQUISITES
  • Understanding of multivariable calculus concepts, particularly partial derivatives.
  • Familiarity with the definition of continuous functions.
  • Knowledge of differentiability and its implications in calculus.
  • Awareness of Clairaut's theorem regarding mixed partial derivatives.
NEXT STEPS
  • Study the implications of differentiability in multivariable functions.
  • Learn about the conditions for continuity of functions in multiple variables.
  • Explore Clairaut's theorem and its applications in higher-order derivatives.
  • Investigate examples of functions with discontinuous partial derivatives.
USEFUL FOR

Students and professionals in mathematics, particularly those studying multivariable calculus, as well as educators seeking to clarify the concepts of continuity and differentiability in functions of several variables.

omri3012
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Hallo,
What is the condition for partial derivatives to be continuous (if I have function f(x,y))?
Thanks,
Omri
 
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Is the partial derivative a continuous function?

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

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

and then apply this to higher order partials {\partial^n y}/{\partial x^n}
 
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".
 

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