= Continuous functions are meagre in L1
The image of $C([0,1])$ in the <L1 space> $L^1(0,1)$ is a <meagre set>. The canonical map $J$ is a <bounded linear operator> from the <Banach space> $C([0,1])$ with its <supremum norm>, since $\|Jf\|_1\leq\|f\|_\infty$. It is not onto: the indicator of $(1/2,1)$ has no <continuous function> representative. Continuity would force values zero throughout $(0,1/2)$ and one throughout $(1/2,1)$, contradicting continuity at $1/2$. The <operator range category dichotomy> therefore proves the claim. Continuous ramps $f_n(t)=\min(1,\max(0,n(t-1/2)))$ also show directly that this image is not closed, since their <L1 norm> distance to that indicator is $1/(2n)$ for $n\geq2$.
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