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Thread: Continuity of sums, products, and quotients of functions

  1. #1
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    Continuity of sums, products, and quotients of functions



    I think this is probably pretty simple. Is this how its done?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by davismj View Post


    I think this is probably pretty simple. Is this how its done?
    Who says that $\displaystyle f$ is continuous at a point means that $\displaystyle \lim_{x\to x_0}f(x)=f(x_0)$. What about $\displaystyle f:\{1\}\mapsto\{1\}$?
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    there exits sequence $\displaystyle (x_n)$ that converges to $\displaystyle z_0$. f and g are continuous at $\displaystyle z_0$ so $\displaystyle f(x_n)\rightarrow f(z_0)$ and $\displaystyle g(x_n)\rightarrow g(z_0)$ when $\displaystyle n\rightarrow \infty$. so $\displaystyle \lim_{n\to \infty}(f+g)(x_n)=\lim_{n\to\infty}[f(x_n)+g(x_n)]=f(z_0)+g(z_0)=(f+g)(z_0)$ same with $\displaystyle fg$..
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