$C([0, 1])$ is not complete with respect to the norm $\lVert f\rVert _1 = \int_0^1 \lvert f (x) \rvert \,dx$

Let $m\leq n$ both natural numbers, then $$\|f_n-f_m\|_1 = \int_0^1 |f_n(x)-f_m(x)|\,\mathrm{d}x $$ $$ = \int_\frac{1}{2}^{\frac{1}{2}+\frac{1}{n}}(n-m)\left(x-\frac{1}{2}\right)\,\mathrm{d}x + \int_{\frac{1}{2}+\frac{1}{n}}^{\frac{1}{2}+\frac{1}{m}}\left(1-m\left(x-\frac{1}{2}\right)\right)\,\mathrm{d}x.$$

Now try to bound these integrals for $n,m\geq N$.

Its correct but some part is still left. You have to show that it is not complete. For this we need to show that limit point is not continuous. Suppose function f(x) is the limit. Observe that it is not right continuous at x=1/2. hence it is not complete