If $f$ and $g$ are continuous maps on a topological space $(X,\tau)$ then $f+g$ is 
If $f$ and $g$ are continuous maps on a topological space $(X,\tau)$ then $f+g$ is
$(a)$ maybe continuous.
$(b)$ may or may not be continuous.
$(c)$ is continuous.
$(d)$ none of these.

I think the answer should be continuous which is option $(c),$ but my book says option $(a)$ i.e. maybe continuous. I don't understand how do I prove this or contradict this. Any help is appreciated thanks. An example demonstrating the fact that if $f,g$ are continuous then $f+g$ may be continuous will also work. Thanks.

Start by thinking about what does $f+g$ really mean. There is a hidden composition of two functions here. Define $h:X\to \mathbb R^2$ (or change $\mathbb R$ for wherever space you are summing) by $h(x) = (f(x),g(x))$ and define $(+): \mathbb R^2 \to \mathbb R$ as $(+)(x,y) = x+y$ (this is essentially what's called polish notation). Now, you have $f+g = (+) \circ h$. In other words, you first compute $f(x), g(x)$ and then you add them.
Now you have split a hard problem into three easier problems:

*

*Is $h$ continuous? (Remember that $f$, $g$ are continuous)

*Is $(+)$ continuous?

*Is the composition of two continuous functions (namely $(+)$ and $h$)  continuous?

