# 2 simliar questions: Am I right?

• Buri
In summary, the first problem involves a function f: R^2 -> R and its directional and partial derivatives at the point (0,0). The directional derivatives exist for all vectors u=(h,k) and the partial derivatives both equal 0. However, since the directional derivative is non-linear, the function is not differentiable at (0,0). Additionally, the function is not continuous at (0,0) due to different values along the line y=x and at the origin.For the second problem, the directional and partial derivatives do not exist at (0,0). However, the function is continuous at (0,0) as shown by using the triangle inequality.
Buri
1. First problem
Let f: R^2 -> R be defined by setting f(0) = 0 and f(x,y) = xy/(x^2 + y^2) if (x,y) is not equal to zero.

For which vectors u not equal to zero, does the directional derivative f'(0;u) exist?

I have that it exists for all u = (h,k) and is equal to hk/(h^2 + k^2)

Do the partial derivatives exist at (0,0)? Yes and they're both equal to 0.

Is f differentiable at 0? No because if it were, its differential at u would equal the f'(0;u), but f'(0;u) is non linear. So its impossible.

Is it continuous at 0? No because arbitrarily close to the origin the function has value 1/2 (this is along the line y = x) but at the origin the function is equal to 0. So it can't be continuous at 0.

2. Second Problem

Its the same thing, just with a differnent function:

f: R^2 -> R f(x,y) = |x| + |y|

I have its direction derivatives don't exist.

I'll write what I did for this one since I'm not exactly sure if its right.

f'(0;u) = lim [t->0] 1/t[f((0,0) + t(h,k)) - f(0,0)] = lim [t->0] 1/t(|th| + |tk|)

See here is where I'm not exactly sure if I'm doing things right. Here I guess I must check that the "right and left" limits exists, but that doesn't really make sense anymore since its a multivariable function. So I basically consider t > 0 and t< 0 and I get |h| + |k| and -|h| - |k|. So they aren't equal and therefore, it doesn't exist.

So directional derivatives at (0,0) don't exist.

So neither do partial derivatives then.

Can't be differentiable. Since all the directional derivatves at 0 don't exist it implies that it is not differerentiable at 0.

Now is it continuous at 0?

I'm trying to show that:

lim [ (x,y) -> (0,0) ] f(x,y) = 0.

I tried proving this directly from the epsilon-delta definition but its not working for me.

I have:

(x² + y²)^(1/2) < delta then (x^2)^(1/2) (y^2)^(1/2) < epsilon

But I can't seem to connect the two. So what I did is I broke up the limit into two parts:

lim [ (x,y) -> (0,0) ] f(x,y) = lim [ (x,y) -> (0,0) ] |x| + lim [ (x,y) -> (0,0) ] |y|

I know the above isn't valid yet since I don't know whether the limit exist, but I'm showing my reasoning.

Now proving that lim [ (x,y) -> (0,0) ] |x| = 0 is basically setting delta equal to epsilon, so its easy. Therefore, the individual limits exists which implies the summation of the limits exist and hence the limit of the summation exists. So it is continuous at 0.

If someone could tell me if I did these right would be awesome!

Thanks for the help!

Anyone? I'm sure you guys all know how to do these lol And besides, I've shown my work! :) So I think I deserve some help :)

Last edited:

Hello! As a fellow scientist, I would be happy to provide some feedback on your work.

For the first problem, your reasoning and calculations seem to be correct. The directional derivatives do exist for all vectors u=(h,k) and the partial derivatives at (0,0) are both equal to 0. However, as you mentioned, since the directional derivative is non-linear, the function is not differentiable at (0,0). Additionally, since the function has different values along the line y=x and at the origin, it is not continuous at (0,0).

For the second problem, your reasoning for the directional derivatives and partial derivatives not existing is correct. However, for the continuity at (0,0), you can use the triangle inequality to show that |f(x,y)| <= |x| + |y|, which means that as (x,y) approaches (0,0), f(x,y) also approaches 0. This shows that the function is continuous at (0,0).

Keep up the good work and happy problem-solving!

## 1. Am I right?

This question is often asked when someone is seeking validation or confirmation for their beliefs or opinions. It implies that the person wants to know if their understanding or perspective on a certain topic is correct.

## 2. How do I know if I am right?

This question is usually asked when someone is unsure about their reasoning or conclusions. They may be seeking guidance or evidence to support their beliefs.

## 3. What makes someone "right"?

This question is frequently asked when discussing subjective topics or opinions. It implies that there is a standard or criteria for being "right" and the person wants to know how to meet those standards.

## 4. Can there be multiple "right" answers?

This question is often posed in situations where there are different perspectives or solutions to a problem. It suggests that there may be more than one correct answer and the person is seeking clarification or confirmation.

## 5. Is there a way to prove that I am right?

This question is commonly asked in scientific or academic settings, where evidence and proof are highly valued. It implies that the person wants to know if there is a method or process to validate their beliefs or conclusions.

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