Interpretation of this Linear Transformation

Question

Why is it necessary to write $F(x)$ instead of just $F$?

Usually I only see $\int_0^xf(t) dt$ where $0≤x≤1$ so why does $F$ need to be a function of $x$?

Can I interpret the mapping $L$ as $L(f)=\int_0^xf(t) dt$ where $0≤x≤1$?

Answer 1

If you want to use only $F$ you may write $$F = \int_0^{\cdot} f(t)\,dt.$$ This notation means that when you give an argument to $F$ it appears at the top of the integral: e.g. $$F(x) = \int_0^x f(t)\,dt, \quad F(x^2) = \int_0^{x^2}f(t)\,dt, \dots$$

Since $x$ is a variable, the integral $\int_0^x f(t)\,dt$ is a function of $x$.

Finally, since $L(f) = F$, you cannot simply write $L(f) = \int_0^x f(t)\,dt$, because the right hand side is $F(x)$, not only $F$. Instead, you may write $$L(f)(x) = F(x) = \int_0^x f(t)\,dt.$$