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ƒ{c} = δ(f) : c ∈ R & f => fourier transform

however i cannot prove this

Here is my attempt:(assume integrals are limits to [-∞,∞])

ƒ{c} = ∫ce

^{-2πft}dt = c∫e

^{-2πft}dt = c∫ƒ{δ(f)}e

^{-2πft}dt

=> cf{f{δ(f)}} => cδ(-f)

this is incorrect, any help would be appreciated thank you.