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- May 17th 2008, 09:01 AMszpengchaoshow its continuity
- May 17th 2008, 01:05 PMflyingsquirrel
Hi

For the continuity of $\displaystyle f$, we know that $\displaystyle \lim_{h\to0}f(x_0+h)=f(x_0)$ and we want $\displaystyle \forall y\in\mathbb{R},\, \lim_{h\to0}f(y+h)=f(y)$. You may try to write $\displaystyle \lim_{h\to 0}f(y+h)=\lim_{h\to0} f(y-x_0+x_0+h)=\ldots$

For the second question, I suggest you first show that $\displaystyle \forall n\in\mathbb{Z}, f(n)=nf(1)$. Then, you'll be able to show that $\displaystyle \forall r\in\mathbb{Q}, f(r)=rf(1)$. As $\displaystyle \mathbb{Q}$ is a dense subset of $\displaystyle \mathbb{R}$ and $\displaystyle f$ is continuous, you'll be able to generalize the result to the real numbers.

Why don't you post text instead of images ? Images are annoying because one can't quote the question he is answering... - May 17th 2008, 06:04 PMThePerfectHacker