So basically, the question is pretty straight forward. But I’m having troubles proving or countering the following statement, because of the constant k.
If $f:D\rightarrow \Bbb R$ is continuous on a topological space $D$ and $k\in \Bbb R$, then $kf$ is continuous on $D$.
Seems to me that I have to use the theorem for multiplication of two products. But $k$ is a constant. So how do I exactly apporach this matter? I know for a fact that any function multiplied by a constant is still continuous if and only if the function itself is continuous.
Let $f:D\to\mathbb{R}$ be a continuous function. Now, you have that $g_\lambda:\mathbb{R}\to \mathbb{R}:x\mapsto \lambda x$ is continuous for every $\lambda\in\mathbb{R}$. Then, consider $h=g_\lambda\circ f:D\to\mathbb{R}$ that is continuous since is the composition of two continuous functions, and $h(x)=\lambda x$.