Show that function $f(x)=gx$ is continuous on $[0,\infty)$

Question

Show that function $f(x)=gx$ is continuous on $[0,\infty)$, where $g$ is constant

Suppose $x>=0$ and $\epsilon>0$. It suffices to show that there exists $\delta>0$ such that for every $y$ in the domain. $|x-y|<\delta =>|gx-gy|<\epsilon$

Case I: $x>0$. Choose $\delta=\epsilon/|g|$

$|x-y|<\delta =>|gx-gy|=|g||x-y|<|g|\delta=|g|\frac{\epsilon}{|g|}<\epsilon$

Case II:$x=0$

$|x-y|<\delta =>|gx-gy|=|g||x-y|=|g||y|<|g|\delta=|g|\frac{\epsilon}{|g|}<\epsilon$

Then is continuous on $[0,\infty)$. Am I ok? Can you help me to improve it? Thanks

Answer 1

You don't neeed to differentiate the case $x=0$ beacuse you can in this case the same reasonning than in II. But you need to treat independently the case $g=0$, because otherwises the quantity $\delta=\epsilon/|g|$ is not defined.