Multiplication by a constant in a definite integral If I'm not mistaken, if $k$ is some constant, $f(x)$ is some real-valued function and $a$ and $b$ $\in \mathbb{R}$  then $\int^b_a k f(x) dx = k \int^b_a f(x) dx$. 
Now, if I know $\int^b_a k f(x) dx= k \int^b_a f(x) dx$ does that mean $k$ is a constant? It feels like it should but it also feels like a strong statement. Any counter-examples or proof?

If $\int_a^b f(x) \neq 0$ then $k$ must be constant, since $k(x) = \frac{\int_a^b kf(y)\,dy}{\int_a^b f(y)\,dy}$ clearly does not depend on $x$.
If $\int_a^b f(x)\,dx = 0$ then the equation holds if $\int_a^b k(x)f(x)\,dx$ also vanishes. An easy such example (with $a\neq b$) is $f(x) = x$, $k(x) = x^2$, $a=-1$, $b=1$.
