(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

study the continuity, directional derivatives, and differentiability of the function f(x,y)=arctan(abs(y)*(y+x^2-1)).

3. The attempt at a solution

the function is obviously continuous in R2 since made of continuous functions.

has directional derivatives everywhere since made of functions that has directional derivatives everywhere.

differentiable everywhere exept for (x,0) and here is my biggest doubt: how do i demonstrate that it isnt differentiable there? if i think about it on a logical level, i know these are point where the function isnt smooth, but how do i demonstrate it?

in (+-1,0) its differentiable because for h=x+-1,k=1:

lim(h,k)->(0,0) arctan(abs(k)*(k+h^2+-2h))/(sqrt(h^2+k^2)) goes obviously to 0.

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