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Is it possible to say that the directional derivatives of a function f at a exists but f is not differentiable at a? If so, why? I cannot get the intuition about it. Could someone please elaborate on this point a little bit?

I am self studying mostly, so I need to discuss these trivial matters with someone :)

Thanks in advance!

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Yes, it's possible, Care Bear's link gives an example.

An explanation as to the why: Directional derivatives gives you the rate of change on any STRAIGHT LINE path. (I.e., the limit along a straight line). In order for a limit to exist in multiple dimensions, it must be the same on any path, not just straight line paths. There's simple examples of functions that all the limits on straight lines are the same, but not on say, parabolas.

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