Showing a linear mapping is continuous (or not) I have three linear mappings:
\begin{equation}t_0(f)=f(t_0)\end{equation}
\begin{equation}I(f)=\int_{0}^{1}f(t)f_0(t)dt\end{equation}
\begin{equation}T(f)=f(t)f_0(t)\end{equation}
and I want to determine whether or not they are continuous on $(C[0,1],\|\centerdot\|_1)$.
I have been trying to prove continuity by showing boundedness, e.g. $|T(f)|\leq M\|f\|_1$, with no success. I also have tried to construct a sequence $f_n$ satisfying $\|f_n\|_1=1$ and $|f_n(t)|\rightarrow\infty$, or find a sequence $f_n$ such that $\int_{0}^{1}f_n(t)dt\rightarrow0$ but $|T(f)|\nrightarrow0$. I have a hard time with counter examples.
I would greatly appreciate any hint or push in the right direction. 

$||f||_1$ is the area under the curve $t \mapsto |f(t)|$. One way of thinking about the first problem is to look for a sequence of simple shapes that have constant area, but the height goes to $\infty$. Rectangles are an obvious choice, except they are not continuous, but this can be fixed easily in many ways.
For the second, consider using Hölder's inequality.
For the third (assuming this is a mapping into $(C[0,1],\|\centerdot\|_1)$), notice that $||Tf||_1 = I(Tf)$, where $I$ is from the second problem.
