Proof that $F$ is continuous in fundamental theorem of calculus
1
I am studying calculus and have arrived at the first FTC. In our book it states that: let $f: [a,b]tomathbb{R}$ be continuous on the interval $[a,b]$ then we define the function $F_f : [a,b]tomathbb{R} : x mapsto F_f(x) := int_a^x f(t),dt$ Then these properties hold: $F_f$ is continuous on $[a,b]$ $F_f$ is differentiable on $]a,b[$ $F_f$ is a primitive for $f$ The proof of the last two properties are quite understandable and also easy to find. However I am confused about the first property (supposedly the most straightforward one) where it states: The continuity of $F_f$ follows immediately from the fact that for every $x,y in [a,b]$ we have: $|F_f(x) - F_f(y)| = int_a^x f(t),dt - int_a^y f(t),dt| = |int_y^x f(t),dt| leq max_{x in [a,b]} |f(x)|cdot |x-y|$ Now here is what I am not sure of...