Kronecker Delta and Constants

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I have a small question and I hope I make this clear.

Would one distribute the Kronecker Delta as follows?

$\sum_{j}^n\ ( C + x_j ) \delta_{ij}$

$= \sum_{j}^n\ ( C\delta_{ij} + x_j\delta_{ij} ) $

$= C + \sum_{j}^n\ x_j\delta_{ij} ) $

I believe the answer to be only : $ \ \ x_i $

But I can not reason why the C term should vanish. Any help is appreciated.