How to show continuous at each point in R^2

[misterau]

Homework Statement

f(x , y) = y^3 + x^3

Calculate the partial derivatives fx and fy and show they are continuous at each point (x,y) ∈ R^2

Homework Equations

A function is continuous on a region R in the xy-plane if it is continuous at each point in R

A function f is continuous at the point (a,b) if
lim f(x,y) = f(a,b)
(x,y)->(a,b)

The Attempt at a Solution

Now calculating the partial derivatives was easy. But not sure how to show show they are continuous at each point (x,y) ∈ R^2?

 

[CompuChip]

There are some theorems which are very useful in cases like this, like
x \mapsto x is continuous
a scalar multiple of a continuous function, as well as the sum of two continuous functions is continuous
a product of two continuous functions is continuous
a quotient of two continuous functions is continuous as long as the denominator is non-zero
the composition of continuous functions is continuous, where defined

There should be such a set of theorems which is proved almost right after the definition of continuous, which we usually apply instead of the definition.

If you want to do the definition, you should calculate the limit, for example
\lim_{(x, y) \to (a, b)} (3 y^2 + x^3)
and show that it is 3b^2 + a^3.

 

[misterau]

There are some theorems which are very useful in cases like this, like
x \mapsto x is continuous
a scalar multiple of a continuous function, as well as the sum of two continuous functions is continuous
a product of two continuous functions is continuous
a quotient of two continuous functions is continuous as long as the denominator is non-zero
the composition of continuous functions is continuous, where defined
.

I am not understanding how this helps me do the problem? If you could show me an example or link to an example I would be thankful.

 

[misterau]

nvm I worked it out. Thanks for the help.

 

[CompuChip]

OK, here is an example:

Let f(x, y) = x + y².

By the first theorem (or a variation on it), (x, y) \mapsto x and (x, y) \mapsto y are continuous. Multiplying the second with itself, (x, y) \mapsto y^2 is continuous. Using the sum rule on the first and the "y-squared" function, f(x, y) is continuous.