Given $f_1,f_2: X \to X$ where $X$ is a metric space and the mappings are continuous and bounded (and therefore Lipschitz). Does it necessarily follow that $$(f_1+f_2)(x) = f_1(x)+f_2(x)$$ for some $x \in X$?
I've seen the property that $(x+y)(t) = x(t) + y(t)$ a couple of times in my life in calculus or in linear algebra sometimes. Are there times when it doesn't hold?
Does the above question hold if $X$ is a normed space?