Problem: Prove using epilson delta definition that $\displaystyle f(x)=kx$ is continuous for any real number $\displaystyle k$.

I have the $\displaystyle k \neq 0$ case. I just had some problems with the $\displaystyle k=0$ case. Here's what I got.

Let $\displaystyle \epsilon > 0$ and choose $\displaystyle \delta=\epsilon$, then $\displaystyle |x-x_{0}| < \delta $ implies that $\displaystyle |x-x_{0}| < \epsilon $ and so $\displaystyle |k||x-x_{0}| < |k|\epsilon $. Then $\displaystyle |kx-kx_{0}| < |k|\epsilon $ and since $\displaystyle k=0$ and $\displaystyle f(x)=kx=0$ and $\displaystyle f(x_{0})=kx_{0}=0$ then $\displaystyle |f(x)-f(x_{0})| < |k|\epsilon $. Since $\displaystyle \epsilon > 0$, then $\displaystyle \epsilon > |k|\epsilon$. So $\displaystyle |f(x)-f(x_{0})| < \epsilon$.

Thoughts? Corrections?