A function with the following properties real valued I am looking for a real valued continuous function $f$ on an interval $[-1,1]$ with the property that $\max_{[-1,1]} |f| = 1$ and $\left |\int_{-1}^{0} f - \int_{0}^{1} f \right|= 2$
Polynomials, trig functions, and exponential functions seem to all fail. I am out of ideas

Such a function $f$ cannot exist. Here is the reason.
Since $|f|\leq 1$, we have $|\int_{-1}^{0} f|\leq \int_{-1}^0|f|\leq 1$. From this we see that $\int_{-1}^{0} f=1$ when $f\equiv 1$ on $[-1,0]$, and  $\int_{-1}^{0} f=-1$ when $f\equiv -1$ on $[-1,0]$. Here we use the fact that $f$ is continuous. 
Similarly, we have $|\int_{0}^{1} f|\leq \int_{0}^1|f|\leq 1$. From this we see that $\int_{0}^{1} f=1$ when $f\equiv 1$ on $[0,1]$, and  $\int_{0}^{1} f=-1$ when $f\equiv -1$ on $[0,1]$. 
Therefore, if $|\int_{-1}^{0} f - \int_{0}^{1} f |= 2$, we must have: 
Case (i). 
$\int_{-1}^{0} f=1\mbox{ and }\int_{0}^{1} f=-1$, or, 
Case (ii). 
$\int_{-1}^{0} f=-1\mbox{ and }\int_{0}^{1} f=1$.
In Case (i), from the above reasoning, we have $f\equiv 1$ on $[-1,0]$ and  $f\equiv -1$ on $[0,1]$, which is a contradiction. In case (ii), again from the above reasoning, we have $f\equiv -1$ on $[-1,0]$ and  $f\equiv 1$ on $[0,1]$, which is again a contradiction.
