I was a bit confused by the definition of integrals (both definite and indefinite) and anti-derivatives. The definition for indefinite integrals is- The indefinite integral of a function x with respect to f(x) is another function g(x) whose derivative is f(x). i.e. g'(x) = f(x) ⇒ Indefinite Integral of f(x) is g(x) In mathematical notation we write- ∫f(x) dx = g(x) + C if and only if g'(x) = f(x) (as d/dx(g(x)+C) = f(x) And the definition of definite integral is- Let f(x) be continuous on [a,b]. If G(x) is continuous on [a,b] and G'(x)=f(x) for all x (a,b), then G is called an anti-derivative of f. We can construct anti-derivatives by integrating. The function F(x)= [itex] \int_a^x f(t) dt [/itex] is an anti-derivative for f since it can be shown that F(x) constructed in this way is continuous on [a,b] and F'(x) = f(x) for all x (a,b). My question is that, firstly, in the definition of an indefinite integral, how can we be sure that g(x) would have to be continuous ∀ x ∈ R? And in the definition of a definite integral, how can we be sure that g(x) would have to be continuous for x ∈ (a,b)? Just because g'(x) = f(x), is this enough to say that g(x) is a continuous function for ∀ x ∈ R and x ∈ (a,b) respectively in the 2 definitions? g'(x) exists means g(x) is a continuous function. But we don't know whether g'(x) exists ∀ x ∈ R. Similarly, [itex] \int_a^x f(t) dt [/itex] is integral in the interval (a,b). So how can we be certain that its anti-derivative would also be valid for only the interval (a,b)? Is this the definition of the indefinite integral that we can't question as to why would the -derivative be valid for only the interval (a,b)?