real analysis - Let $f(x)$ be Riemann-integrable, $F(x)=int_a^x f(t)dt.$ Then $F(x)$ is differentiable, and $F'(x)=f(x)$ almost everywhere.

What is wrong with the following statement:

Let $f(x)$ be Riemann-integrable, $$F(x)=\int_a^x f(t)dt.$$ Then $F(x)$ is differentiable, and $F'(x)=f(x)$ almost everywhere.

I think the statement is true. Because $f(x)$ be Riemann-integrable, it's continuous almost everywhere. Then by the Second Fundamental Theorem of Calculus, $F(x)$ is is differentiable almost everywhere, and $F'(x)=f(x)$ almost everywhere.