# Continuity and differentiability in two variables

• wavingerwin
In summary, if a function ##f(x,y)## is independently continuous in ##x## and ##y##, it does not necessarily mean that it is continuous in both variables. This can be seen in the counterexamples provided in Ch 9, section 1 of "Counterexamples in Analysis". Additionally, being independently differentiable in ##x## and ##y## does not guarantee differentiability in both variables.
wavingerwin
Hi

If the function ##f(x,y)## is independently continuous in ##x## and ##y##, i.e.
$f(x+d_x,y) = f(x,y) + \Delta_xd_x + O(d_x^2)$ and $f(x,y+d_y) = f(x,y) + \Delta_yd_y + O(d_y^2)$
for some finite ##\Delta_x##, ##\Delta_y##, and small ##\delta_x##, ##\delta_x##,

does it mean that it is continuous in both?
$f(x+d_x,y+d_y) = f(x,y) + \Delta_xd_x +\Delta_yd_y+O(d_x^2,d_y^2)$

How about differentiability? (if the function is independently differentiable in ##x## and ##y##, is it differentiable in both ##x## and ##y##?)

Cheers
wavingerwin

wavingerwin said:
does it mean that it is continuous in both?
By "is continuous in both", do you just mean "is continuous"? Then no.. See Ch 9, section 1 of Counterexamples In Analysis (p 115 of the book, p 140 of the PDF) http://www.kryakin.org/am2/_Olmsted.pdf

wavingerwin

## 1. What is the difference between continuity and differentiability in two variables?

Continuity in two variables means that a function has no sudden jumps or breaks in its graph when both variables are changed together. Differentiability in two variables means that a function has a well-defined slope at every point when both variables are changed together.

## 2. How can I determine if a function is continuous in two variables?

A function is continuous in two variables if it is continuous in both variables individually and at the point where they intersect. This means that the limit of the function as both variables approach the point must exist and be equal to the value of the function at that point.

## 3. Can a function be differentiable but not continuous in two variables?

No, a function cannot be differentiable in two variables if it is not continuous. This is because differentiability requires the existence of a limit at a point, and if the function is not continuous at that point, the limit does not exist.

## 4. What is the relationship between continuity and differentiability in two variables?

Continuity is a necessary condition for differentiability in two variables. In other words, a function must be continuous in order to be differentiable. However, being continuous does not necessarily mean that a function is differentiable.

## 5. How can I check if a function is differentiable in two variables?

To check if a function is differentiable in two variables, you can use the partial derivative test. This involves taking the partial derivatives of the function with respect to each variable and checking if they are continuous. If they are, then the function is differentiable in two variables. Additionally, you can also check if the limit of the difference quotient exists at a point, which is equivalent to checking if the function is continuous at that point.

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