Zero to the Power of Zero/Binomial Theorem

Example of Zero to the Power of Zero
Consider the real polynomial function:


 * $y = \left({x + c}\right)^n$

for $n \in \N, c \in \R$.

By the binomial theorem, $y$ contains a term of the form:


 * $\dbinom n n x^{n - n} c^n$

If we did not define $0^0 = 1$, $y$ would have a discontinuity at $x = 0$.

This would contradict the theorem that a polynomial is continuous on the entire real line.