Calculus prove that $F(x)= \int_{-\infty}^xf(t)dt$ is continuous Let $f: \mathbb{R} \to \mathbb{R}$ be a function for which  $\int_{-\infty}^xf(t)dt$ converges for every $x \in \mathbb{R}$.
Consider the function $F$ defined as $F(x)= \int_{-\infty}^xf(t)dt$ for every $x \in \mathbb{R}$.
Prove that $F$ is continuous in $\mathbb{R}$.
My attempt:
Assuming towards contradiction that $F$ is not continuous, so there exists a $x_0$ such that $F(x_0) \ne \lim_{x \to x_0}F(x_0)$.
But: $F(x_0)=\int_{-\infty}^{x_0}f(t)dt=L$
$\lim_{x \to x_0} F(x_0)=\lim_{x \to x_0}\int_{-\infty}^{x_0}f(t)dt=\lim_{x \to x_0}L=L$
So $F(x_0) = \lim_{x \to x_0}F(x_0)$ - contradiction.
Is that correct?
Thanks!
 A: You need to prove that $F$ is continuous and you are just stating that the limit is what it should be. Consider the following proof instead. Let $x_{0}\in\mathbb{R}$ be fixed and let $\epsilon>0$ be given. Then, take $x>x_{0}$ WLOG and note that:
\begin{equation*} |F(x)-F(x_{0})|=\left\vert\int_{-\infty}^{x}f(t)dt-\int_{-\infty}^{x_{0}}f(t)dt\right\vert=\left\vert\int_{x_{0}}^{x}f(t)dt\right\vert\leq\int_{x_{0}}^{x}|f(t)|dt\leq M(x-x_{0}) \end{equation*}
where $M=\sup_{t\in[x_{0}-1,x_{0}+1]}|f(t)|$ (this supremum exists because $f$ is Riemann-integrable and therefore bounded on the compact interval $[x_{0}-1,x_{0}+1]$). The continuity of $F$ at $x_{0}$ then follows by taking $x\to x_{0}$ so the above quantity is bounded above by $M\epsilon$ and there is an analogous argument for the $x<x_{0}$ case.
A: It's not quite correct. There's one actual issue with your proof, together with another problem that, while not incorrect, isn't optimal.
The actual issue is that your definition of continuity isn't quite correct. It's so small it might appear to be a typo, but your usage of it in your proof indicates it's not. It looks like you've defined continuity to be that $F$ is continuous iff $F(x_0)=\lim_{x\to x_0}F(x_0)$, however, if this were the true definition then every function would be continuous. The actual definition of continuity is $F(x_0)=\lim_{x\to x_0}F(x)$, without the subscript $0$ for the $F$ in the limit.
The second problem is look at the form of your argument: You're tasked with showing two things are equal. You begin by assuming they are not equal, show they are in fact equal, which is a contradiction, so your assumption that they were not equal must have been false, therefore they are equal. If this sounds long and over-worked, it's because it is. You were able to generate a contradiction by showing they were in fact equal, which is the very thing you intended to prove in the first place. You can transform this proof by contradiction into a direct proof by erasing the first and second to last lines, yielding a proof by way of showing that they are equal, therefore they are equal.
