Continuity of sums, products, and quotients of functions

**davismj** · January 27th 2010, 09:27 AM

I think this is probably pretty simple. Is this how its done?

**Drexel28** · January 27th 2010, 01:00 PM

Originally Posted by davismj

I think this is probably pretty simple. Is this how its done?

Who says that $f$ is continuous at a point means that $\lim_{x\to x_0}f(x)=f(x_0)$ . What about $f:\{1\}\mapsto\{1\}$ ?

**Julius** · January 28th 2010, 11:02 AM

there exits sequence $(x_n)$ that converges to $z_0$ . f and g are continuous at $z_0$ so $f(x_n)\rightarrow f(z_0)$ and $g(x_n)\rightarrow g(z_0)$ when $n\rightarrow \infty$ . so $\lim_{n\to \infty}(f+g)(x_n)=\lim_{n\to\infty}[f(x_n)+g(x_n)]=f(z_0)+g(z_0)=(f+g)(z_0)$ same with $fg$ ..

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