Discrete isn't continuous, so technically the assumption of the t-test is not met, and that's that. However, the t-test is fairly robust and having $N=48$ with equal groups is a decent sample, so it might be OK. After all, in practice all data are discrete at some level because we don't record data to infinite decimal places.
I would look (i.e., plot; cf. R - QQPlot: how to see whether data are normally distributed & How to interpret a QQ plot) at your data to see if they appear reasonably normal and meet the other assumptions (e.g., homoscedasticity). I would not run formal tests for normality (cf. Is normality testing 'essentially useless'?). If they look 'close enough', you're probably OK.
On the other hand, it is generally considered sub-optimal to follow the procedure 'check assumptions, if OK, run test'. If you think this might be a problem, you can just go straight to a more robust option. (Cf. How to choose between t-test or non-parametric test e.g. Wilcoxon in small samples.)