I had this question on an old Hw. Here is the solution i gave for it. The function was
instead of what you posted but its the same thing, switch the components in the proof.
As for directional derivatives.
Show that exists.
"show that is continuous at (0,0), and that all directional derivatives exist at (0,0)". how would i go about showing continuity? the denominator is 0 and i guess i have to do something about that? i can't seem to cancel it down. i'm really stuck for ideas.
I had this question on an old Hw. Here is the solution i gave for it. The function was
instead of what you posted but its the same thing, switch the components in the proof.
As for directional derivatives.
Show that exists.
hi,
i think that answer you attached is a little too advanced for what we're doing. there's a lot of notation we haven't covered. thanks for your help, but is it possible there's a simpler way of looking at this? i totally don't follow, the x^j, E^2 etc. i follow that showing the limit using the epsilon delta definition can show continuity. perhaps what you've done is the same thing but i'm just not understanding it.
i have that:
for any epsilon > 0, we can find delta > 0 such that < epsilon. whenever < delta. is that in any way analagous to what you've shown here?
and then show that L = 0, show that the function at (0,0) = 0, and thus that implies contunuity?
Have you all forgotten that a function is only continuous if the limit exists at that point and IS EQUAL TO THE FUNCTION VALUE at that point? This function is undefined at that point, so the function is clearly NOT continuous there.
However the function IS continuous at (0, 0).
Take any non zero vector v =
the directional derivative at (0,0) is defined as
whose limit exists since v is non zero and thus the directional derivative exists.
so i get it, you need to show that the condition for existance is applicable to all value of v. therefore for any choice of v, the limit exists, i.e. the directional derivative exists, this is the concept?
excellent, thanks very much. finally, one part of the question i forgot to mention: how can i show does not exist? should i look at approaching along different directions? will that happen?
to show indifferentiably, is it sufficient to show the limit in any direction is nonzero??
hi,
i follow, but polar coords are not assessed on my course. i believe i am supposed to show it without any such conversion (even if it is totally valid). is there any way to do it sticking with xs and ys?