I have read the question posed in Non degenerated critical points and checking the proof given in the accepted answer (I can not use Morse lemma) a question arose in my mind. The idea of the proof is clear althought I have a question on it. Suppose you want to prove that if the determinant of the Hessian matrix at a critical point (that is supposed to be $0$) is not $0$ then the critical point is isolated. Suppose by contradiction that the point is not isolated. The author of the answer consider all curves $\gamma$ (and then straight lines) through the point and then correcly state that $$ (f|_{\gamma})=(\gamma'(t))^T\,H(\gamma'(t)) $$ that in case of a straight line becomes $$ (f|_{\gamma})''=v^TH\,v $$ I do not understant why in this case necessarily $f$ must have a local extremum and then the zero of $f'$ has to be isolated.
Any help will be very appreciated.