If $f(x)$ is continuous at the point $x=0$ and for all real numbers $x$ and $y$, the function $f(x+y)= f(x) + f(y)$. Show that $f$ is continuous for all values of $x$.
Not sure where to begin this problem. Any info on it would be appreciated.
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1Add some context. What is the domain, what is the range of the function? What is the definition of continuity you're using? Are you familiar with many equivalent definitions? – Just dropped in May 10 '18 at 22:15
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For every $x \in \mathbb{R}$, we have $$|f(x +h) -f(x)| = \left|\left(f(x) +f(h)\right) -\left(f(x) +f(0)\right)\right| = |f(h) -f(0)| \underset{h \rightarrow 0}{\longrightarrow} 0$$ since $f$ is continuous at $0$.
This shows that $f$ is continuous on $\mathbb{R}$.
v_lentin
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Note that $f(0)=f(0+0)=f(0)+f(0)=2f(0)$
Thus $f(0)=0$
$$\lim_ {h\to 0} f(x+h)-f(x)=\lim_ {h\to 0} f(h)=f(0)=0$$
Therefore $f(x)$ is continuous at $x$
Mohammad Riazi-Kermani
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