<s>[INST] Show a function is continuous on $\mathbb{R}$ by any method Let $f(x)=\frac{Kx}{K^2+x^2}$ where K is some constant, show this is continuous on $\mathbb{R}$.
Here are my scratch work in looking for a delta. let $x,y\in \mathbb{R} $ WTS $|\frac{Kx}{K^2+x^2}-\frac{Ky}{K^2+y^2}|<\epsilon,\forall \epsilon>0$ whenever $|x-y|<\delta$. So, $|\frac{Kx}{K^2+x^2}-\frac{Ky}{K^2+y^2}|=|K||\frac{x(K^2+y^2)-y(K^2+x^2)}{(K^2+y^2)(K^2+x^2)}|\leq |\frac{xK^2-yK^2+xy^2-yx^2}{(K^2+y^2)(K^2+x^2)}|\leq|\frac{xK^2-yK^2+xy^2}{(K^2+y^2)(K^2+x^2)}|$ I'm a bit stuck here on how to relate this inequality to $|x-y|$. Can someone help out? Thanks!
 [/INST] Since you are asking a solution by any method, you can try to use the limit definition of continuity. Recall that a function $f$ is continuous at a point $c$ of its domain if $$\lim_{x\rightarrow c} f(x) = f(c)$$
So for any $c\in \mathbb{R}$ we have $\lim_{x\rightarrow c}Kx = Kc$ and $\lim_{x\rightarrow c} K^2+x^2 = K^2+c^2 \neq 0$. So we have:$$ 
    f(c) = \frac{Kc}{K^2+c^2} = \frac{\lim_{x\rightarrow c}Kx}{\lim_{x\rightarrow c}K^2+c^2} = \lim_{x\rightarrow c}\frac{Kx}{K^2+x^2} = \lim_{x\rightarrow c} f(x)
$$
</s>