Let f(x + y) = f(x) + f(y) for all x, y in R. | vedclass

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(D) The given equation is Cauchy's functional equation $f(x + y) = f(x) + f(y)$.If $f$ is continuous at even one point,then $f(x) = cx$ for some const

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$(B)$ or $(C)$ both

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(D) The given equation is Cauchy's functional equation $f(x + y) = f(x) + f(y)$.If $f$ is continuous at even one point,then $f(x) = cx$ for some constant $c$,which is continuous everywhere.However,if we do not assume continuity,there exist non-linear solutions (using Hamel basis) that are discontinuous everywhere.Thus,$f(x)$ may be continuous (e.g.,$f(x) = cx$) or $f(x)$ may be discontinuous (non-linear solutions).Therefore,both $(B)$ and $(C)$ are possible.

Let $f(x + y) = f(x) + f(y)$ for all $x, y \in R.$ Then:

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