Fourier transform of product of Dirac delta function and other function

Question

what is Fourier transform of $$(\delta(x)f(x)) $$? $$\tau(f(x)\delta(x))=\tau(f(0)\delta(x))=\tau(f(0))\bigstar\tau(\delta(x))$$ $$\tau(\delta(x))=1$$ what is $$\tau(f(0))??$$ Is the answer the following? $$\tau(f(x)\delta(x))=f(0)$$

Answer 1

Use the fact that $$g(x_{0})=\int_{-\infty}^{\infty}g(x)\delta(x-x_{0})dx$$

With the definition of the Fourier transform: $$\mathcal F\{f(x)\}(k)=\int_{-\infty}^\infty f(x)e^{-ikx}dx$$

Hint: $g(x)=f(x)e^{-ikx}$