Let $Tf(x)=\int_{[0,x]}fdm$ for $0 \leq x \leq 1$. Prove that $C_0((0,1])=\{f \in C[0,1] : f(0)=0\}$ is a closed subspace of $C[0,1]$

Question

Let $Tf(x)=\int_{[0,x]}fdm$ for $0 \leq x \leq 1$.

Prove that $C_0((0,1])=\{f \in C[0,1] : f(0)=0\}$ is a closed subspace of $C[0,1]$

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I need a bit of help with this one. I was thinking of proving that the complement is open, or maybe showing that given any convergent sequence in $C_0((0,1])$ its limit has to be in $C_0((0,1])$.... Could use some help though!! THanks!