Determine whether each of the following real functions of two real variables is continuous:

[desert sentinel]

please help me

[inland geyser]

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[desert sentinel]

i have no idea how to proceed

i just need help with this one so i can do the rest of the exercises

whoever comes along pls ping me

[wanton sequoia]

Consider f(t,t) when t approaches 0

[desert sentinel]

wdym

[wanton sequoia]

Nvm

It doesn’t work

Here if toi want to check if it’s indeed continuous

[desert sentinel]

cause in the single variable limit continuity you do this thing like f(0) = lim x->f(0)

etc

can you please write it?

[desert sentinel]

wdym

[wanton sequoia]

Basically you want to see if when (x,y) approches 0 if f(x,y)

so for example considering all norms are equivalent in a finite dimensional space you check if when ll(x,y)ll approches 0 then so does | f(x,y)| with ll.ll denoting the euclidien norm

[desert sentinel]

i dont understand

[wanton sequoia]

Do you know what a norm is ?

[desert sentinel]

im in a non english speaking country

lol

[wanton sequoia]

Oh okay my bad

[wanton sequoia]

Okay, so basically you want to prove ( or disprove) that lim(x,y)—>(0,0) f(x,y)=0

here take f(t,t^2)

You will see that f(t,t^2) is a non zéro constant

So the function can’t be continuous at (0,0) because else it would tend to 0 regardless of the path taken

[desert sentinel]

ok but how do i calculate that

can you write that

its realy hard to understand on discord @wanton sequoia

[wanton sequoia]

Well what’s f(t,t^2) for a fixed non zéro t?

Just write it

[timid palm]

$f$ is continuous at $(0, 0)$ if and only if:

For every sequence $((u_n, v_n)){n \in \mathbb{N}}$ that converges to $(0, 0)$, the sequence $(f(u_n, v_n)){n \in \mathbb{N}}$ also converges to $f(0, 0)$.

[pallid maple]

Rion

[timid palm]

However, as Rotor suggests, the sequence $u_n = \frac{1}{n}$ and $v_n = \frac{1}{n^2}$ will show you that the second property does not hold