I'm working through *Undergraduate Topology* by Kasriel, and the author asks the following:

> 
> Let $X$ be the set of all continuous, real-valued functions defined on
>  $\left[a,b\right]$. Does \begin{align}
>  \Gamma=\left\{\left(f,\int\_a^bf\right):f\in X\right\},\tag{1}
>  \end{align} represent a function, or further, a 1-1 function?
> 
> 
> 

So far I have determined that it cannot be 1-1 because on some interval $\left[a,b\right]$, $\int\_a^b f$ always results in the same value as it depends only on the endpoints. Hence, the range of this set will be the singleton $\left\{\int\_a^b f\right\}$. Now, what of the domain? If $f$ is monotonic, then I'm guessing this set could be defined on all possible $\left[a,b\right]\subset\mathbb{R}$. If, however, it is periodic with period $L$, like $\cos$ or $\sin$, then it would only be a function so long as throughout the interval, some subset of that period, the function is monotonic.

Does this sound right? I feel like there's a flaw somewhere in my logic.