Can a function $f$ have an antiderivative even though its indefinite integral $F(x) = \int_{a}^{x} f(t)\ dt$ is not one? The fundamental theorem of calculus states that if $f:[a,b] \to \mathbb{R}$ is integrable and $F(x) = \int_{a}^{x} f(t)\ dt$, then $F'(x) = f(x)$ at every point $x$ at which $f$ is continuous. This means that if $f$ is integrable, $F'(x) = f(x)$ almost everywhere. If $f$ is continuous on $[a,b]$, then $F$ is an antiderivative of $f$, since $F'(x) = f(x)$ holds for all $x \in [a,b]$. 
But what if $f$ has a discontinuity at some $x \in [a,b]$? In this case, it is not necessarily true that $F'(x) = f(x)$, and so we cannot necessarily conclude that $F$ is an antiderivative of $f$. Does this mean that there is no antiderivative of $f$? Is it possible for $f$ to have an antiderivative but the indefinite integral $F$ is not an antiderivative of $f$?
I know that if $f$ has a jump discontinuity, then $f$ can have no antiderivative (since the derivative of a function must satisfy the intermediate value property), but what if we have some other type of discontinuity?

Assume $f:[a, b] \to\mathbb {R} $ is Riemann integrable on $[a, b] $ and possesses an anti-derivative $g:[a, b] \to\mathbb {R} $ on $[a, b] $ ie $g'(x) =f(x) \, \forall x\in[a, b] $. Then by Fundamental Theorem of Calculus we have $$g(x) =g(a) +\int_{a} ^{x} f(t) \, dt=g(a) +F(x) $$ It follows that $$F'(x) =g'(x) =f(x) $$ for all $x\in[a, b] $.
Therefore the indefinite integral $F:[a, b] \to\mathbb {R} $ defined by $$F(x) =\int_{a} ^{x} f(t) \, dt$$ also acts as an anti-derivative of $f$ on $[a, b] $.
Without symbols we can summarize as

If a Riemann integrable function possesses an anti-derivative over some closed interval then its indefinite integral also acts an anti-derivative over the same interval. 

In short the answer to your question in bold is NO if we assume Riemann integrability of $f$. However not every Riemann integrable function possesses an anti-derivative. Continuity is only a sufficient for existence of anti-derivative and jump discontinuity is only a sufficient condition for non-existence of anti-derivative. 
