Determine if the following is a homomorphism and/or a isomorphism

Question

Let F be the additive group of all continuous functions mapping $\mathbb{R}$ to $\mathbb{R}$. Let $\mathbb{R}$ be the additive group of real numbers. Define $\phi:$ F $\rightarrow \mathbb{R}$ by $$\phi(f)= \int_{0}^{4} f(x) dx$$

I have not worked with an integral before to determine a homomorphism. Any help is appreciated!

Answer 1

It just follows by the fact that the integral is additive:

$$ \begin{aligned} \phi(f+g)&:=\int_{0}^{4}(f+g)(x)dx\\ &:=\int_{0}^{4}(f(x)+g(x))dx\\ &=\int_{0}^{4}f(x)dx+\int_{0}^{4}g(x)dx\qquad\text{by the additivity of the integral}\\ &:=\phi(f)+\phi(g). \end{aligned} $$

Although $\phi$ is surjective, it is not injective, and hence, is not an isomorphism.

Answer 2

To prove that $f$ is not injective, notice that $$\int_0^4 (2x-4)dx=(0-2)^2-(4-2)^2=0$$ and $$\int_0^4 0dx=0,$$ but $g \not\equiv 0$, where $g(x):=2x-4$.