$f_n:[0,1]\to [0,1]$ be a continuous function and let $f:[0,1]\to [0,1]$ be defined by $f(x)=\operatorname{lim\;sup}\limits_{n\rightarrow\infty}\; f_n(x)$ Then $f$ is
continuous and measurable
continuous, but need not be measurable
measurable, but need not be continuous
need not be measurable or continuous.
I guess $3$ is correct, but I'm not able to prove it.