I'm a grad student who's having a little trouble proving this sort of problem in my intermediate analysis course.
I have $f(x)$ defined as a Riemann integrable function in R defined on some inclusive interval [b,c]. $$F(x)=\int_{b}^{x} f(w) dw $$ I need to DIRECTLY show that my function, $f(x)$ is continuous on $(b , c) $ Is there any way to do this without using the fundamental of calculus?
You don't need the fundamental theorem of calculus here, this is an easier statement. If $f$ is Riemann integrable then it is bounded. Let $M=\sup\{|f(t)|: t\in [b,c]\}$. Then for $x\in (b,c)$ and a sufficiently small $h$:
$|F(x+h)-F(x)|=|\int_b^{x+h} f(t)dt-\int_b^x f(t)dt|=|\int_x^{x+h} f(t)dt|\leq\int_x^{x+h} |f(t)|dt\leq\int_x^{x+h} Mdt=M(x+h-x)=Mh\to 0$
Which proves continuity at the point $x$. Similarly you can prove one sided continuity at the corners of the interval.