# If multi valued functions aren't functions how can they be differentiated and integrated

I recently learned that relations like $$f(x)=\sqrt{x}$$ and $$f(x)=\arcsin(x)$$ are not actually functions but multivalued functions, since they take multiple outputs for a single input. So how come we can write formulas such as $$\frac{\mathrm{d}}{\mathrm{d}x} \big(\arcsin(x) \big)=\frac{1}{\sqrt{1-x^2}}$$, as far as I know, the derivative is defined as $$f'(x)=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$, which is defined for functions. We can also write $$\int_a ^b \sqrt{x}\mathrm{d}x$$, and it is well defined. My question is, what's going on? How can we apply a limit to something that isn't a function?

The confusion happens when you do not distinguish between the functions $$y= \sqrt x$$ which is the positive branch of $$x=y^2$$ and the negative branch which is $$y=-\sqrt x$$
For example $$\sqrt {25}= 5$$ and $$-\sqrt {25}= -5$$
For $$arcsin(x)$$ you pick the branch which satisfies $$-\pi /2 \le \arcsin (x)\le \pi/2$$ and that is well-defined.
For example $$\arcsin ( 1/2)=\pi/6$$