First fundamental theorem of calculus: $$g(x) = \int_a^xf(t)dt$$ then $$g'(x) = f(x)$$
But how does this guarantee the existence of antiderivatives of functions? Tutorials always state it does, but never explain why.
Also, once we know it guarantees the existence of antiderivatives of functions, does this mean we can always use the second fundamental theorem (where you split a definite integral into two indefinite ones)? Is that the key purpose to realise here?