Tuesday, 19 April 2016

real analysis - Example for a continuous function that has directional derivative at every point but not differentiable at the origin



Can we find a function f:RnR that such that f is continuous and vf(p) exists for all pRn and vRn. But f is not differentiable at 0?




Is such function f exists?



Here give a example that has directional derivative everywhere, but it's not continuous at the origin.


Answer



Consider the polar coordinates  (r Φ)  in  R2.  Function f:R2R,  which in polar coordinates is given by:



f(r Φ) := rsin(3Φ)



is continuous everywhere, is infinitely differentiable outside the origin  O,  and  f  has all directional derivative at  O,  but  f  is not differentiable at  O.



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