Can i have a hessian matrix which has some elements that equal zero?

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I read somewhere that in order to have a hessian matrix and be able to conduct further investigation to it, all the variables must be differentiable and they must all exist. Now i have come up with a hessian matrix in which i have 3 zeros. One in the diagonal and two in the off-diagonal part. I want to make sure that this matrix satisfies the above-mentined criterion and that i can continue with the manipulation of the matrix. Thanks in advance