Sunday, 27 July 2014

calculus - continuity at zero implies continuity everywhere


Suppose f(x) is defined on the entire line and continuous at the
origin with the property that f(α+β)=f(α)+f(β)
where α,βR. Prove that f(x) must be
continuous at every point x=a.





Try:



First notice that f(0)=f(0+0)=f(0)+f(0)=2f(0)f(0)=0. Since f continuous at zero, then lim. Next, let a be arbitary real number. Then,



\lim_{ x \to a} f(x) = \lim_{x \to a} f(x-a)+f(a)=_{h = x-a} \lim_{h \to 0} f(h) + f(a) = 0 + f(a)=f(a)



So f is continuous everywhere.




Is this a correct solution?

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