we are given

$$

U(x,t)=\int_{-\infty}^xu(x,t)dx

$$

and are to prove that $U$ satisfies $U_t=kU_{xx}$.

The proof given is "one can see easily that as $x\rightarrow -\infty$ and that therefore $U$ and all its derivatives have to be zero". But the integral over any function with the upper bound approaching the lower bound goes to zero!

For instance I take the function $f(x)=x^5$, which obviously does not satisfy the heat equation for $x\neq 0$. Then isn't

$$lim_{x\rightarrow -\infty}\int_{-\infty}^{x}x^5dx=0$$, and according to this "proof" $\int_{-\infty}^{x}x^5dx$ would satisfy the heat equation?

This does not make sense to me...

Thanks!