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(B, C) Given the functional equation $f(x+y)=f(x)+f(y)$,which is Cauchy's functional equation.Setting $x=0, y=0$,we get $f(0)=f(0)+f(0)$,which implies

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$f(x)$ is continuous $\forall x \in R$

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(B, C) Given the functional equation $f(x+y)=f(x)+f(y)$,which is Cauchy's functional equation.Setting $x=0, y=0$,we get $f(0)=f(0)+f(0)$,which implies $f(0)=0$.Since $f(x)$ is differentiable at $x=0$,we have $f^{\prime}(0) = \lim_{h \rightarrow 0} \frac{f(0+h)-f(0)}{h} = \lim_{h \rightarrow 0} \frac{f(h)}{h} = k$ (where $k$ is a constant).Now,for any $x \in R$,the derivative $f^{\prime}(x)$ is given by:$f^{\prime}(x) = \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} = \lim_{h \rightarrow 0} \frac{f(x)+f(h)-f(x)}{h} = \lim_{h \rightarrow 0} \frac{f(h)}{h} = f^{\prime}(0) = k$.Since $f^{\prime}(x) = k$ for all $x \in R$,$f^{\prime}(x)$ is a constant function.Integrating $f^{\prime}(x) = k$,we get $f(x) = kx + C$. Since $f(0)=0$,we have $C=0$,so $f(x) = kx$.$A$ linear function $f(x) = kx$ is continuous and differentiable for all $x \in R$.Thus,statements $(B)$ and $(C)$ are true.

Let $f: R \rightarrow R$ be a function such that $f(x+y)=f(x)+f(y), \forall x, y \in R$. If $f(x)$ is differentiable at $x=0$,then which of the following statements are true?

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