Why is the cross-correlation an integral?

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The cross-correlation of continuous $f,g$ is:

$$(f \star g)(\tau)=\int_{-\infty}^{\infty}f^*(t)g(t+\tau)dt$$

Why is it an integral?

Why doesn't

The cross-correlation of continuous $f,g$ is:

$$(f \star g)(\tau)=f^*(t)g(t+\tau)$$ suffice?