Is the Ito integral with respect to a brownian motion continuous? According to the book "Stochastic differential equations and applications" by Oksendal we merely know that the Ito integral has continous modification. 
However in Karatsas and Shevre book it is stated that we obtain a continuous process by defult(page 146).
Are the latter authors a bit sloppy or is this really true?
 A: All of the authors are equally correct. The reason is that their constructions are simply different (though against Brownian motion, which is the only integrator Oksendal considers, the resulting continuous processes turn out to be the same). Let me summarise the constructions and point out the differences. Since I'm summarising two different constructions this post will be slightly long, but hopefully the details are already familiar to you.
In both cases, the relevant stochastic integral for simple integrands is defined and easily seen to be continuous and so I take this for granted (note though that even these integrals are slightly different).

First let me discuss the approach taken by Karatzas and Shreve. The page you refer to concerns the integral against a continuous local martingale but to get this they just use localisation and the construction against an continuous $L^2$-bounded martingale, so I'll discuss the construction only in this restricted setting since localisation doesn't affect continuity. Let $\mathcal{M}_c^2$ be the space of continuous $L^2$-bounded martingales. Karatzas and Shreve have earlier shown that this a complete metric space for the translation invariant metric
$$d(X,Y) := \sum_{n \geq 0} \frac{\|X-Y\|_n \wedge 1}{2^n}$$
where $\|X\|_n = \mathbb{E}[X_n^2]^{\frac12}$. 
Fix $M \in \mathcal{M}_c^2$. We've already defined the integral against $M$ as a map $I: \mathcal{S} \to \mathcal{M}_c^2$ (where $\mathcal{S}$ is the space of simple integrands). In general, we hope to integrate a progressively measurable integrand $F$ such that $[F]_n = \mathbb{E}\bigg[\int_0^n F_s^2 d \langle M \rangle_s \bigg]^{\frac12} < \infty$ for each $n$. The space $L^2(M)$ of all such $F$ is a metric space for the translation invariant metric
$$d^\prime(F,G) := \sum_{n \geq 0} \frac{[F-G]_n \wedge 1}{2^n} \to 0.$$
They proceed in several steps.
First, they show that there are simple processes $F^n$ such that as $n \to \infty$, $d^\prime(F,F^n) \to 0$. Second, they show that for simple processes we have an Ito isometry; which they apply in the form $$d(I(F^n),I(F^m)) = d^\prime(F^n,F^m).$$ This shows that $I(F^n)$ is a Cauchy sequence in the complete metric space $\mathcal{M}_c^2$ and so converges to some element of $\mathcal{M}_c^2$ which they call $I(F)$. In particular, it follows from this construction that $I(F)$ is automatically continuous because it was constructed as an element of a space of continuous processes. Note also that they construct the Ito integral as an entire process rather than constructing it at each fixed time.

Now for Oksendal's construction, which only constructs the integral against Brownian motion. This construction is less sophisticated but has the advantage of getting an integral against Brownian motion with less machinery - especially since $B$ is not an $L^2$-bounded martingale!
The first thing to notice is that instead of trying to construct the Ito integral as a whole process as in Karatzas and Shreve, Oksendal constructs the Ito integral for a fixed time interval $[S,T]$. Already we have lost hope that the construction will give us a continuous process since for each $t$, $\int_0^t F_s dB_s$ has been constructed separately and so we can't reasonably hope to get continuity of these objects as a family without making a modification of the process.
From here on, the approach is fairly similar. Oksendal shows that for progressively measurable $F$ such that $\mathbb{E}\bigg[\int_S^T F_s^2 ds \bigg] < \infty$ there are simple integrands $F^n$ such that 
$$\mathbb{E}\bigg[\int_S^T (F-F^n)^2 ds \bigg] \to 0$$
as $n \to \infty$; this is the direct analogue of what we did earlier but at fixed times. Then he uses an Ito isometry to conclude that $I(F^n) = \int_S^T F_s^n dB_s$ is a Cauchy sequence in $L^2(\mathbb{P})$ and hence converges, this time in $L^2(\mathbb{P})$ to some limit which he calls $I(F) = \int_S^T F_s dB_s$. Again we lost hope of immediately getting continuity since this integral is defined as an element of $L^2(\mathbb{P})$ and hence only up to $\mathbb{P}$-a.e. for each fixed $S,T$ (as opposed to defined at all times $[S,T]$ up to $\mathbb{P}$-a.e. Notice the order swap!).

So in summary, Oksendal has constructed the Ito integral as an element of $L^2(\mathbb{P})$ at fixed times and not as a process and so needs to take all of these elements as a collection to form a process and show that that process has a continuous modification. Karatzas and Shreve use a bit more machinery but construct the whole Ito integral as a continuous $L^2$-bounded process in one go for a much wider class of integrators.
