Prove that if $f(x)$ is continuous on $[a,b]$ and we have a point $e$ such that $a\lt e\lt b$, then \begin{eqnarray}&&\int_a^b\,f(x)dx\\&=&\int_a^e\,f(x)dx+\int_e^b\,f(x)dx.\end{eqnarray}

Problem: 

Prove that if $f(x)$ is continuous on $[a,b]$ and we have a point $e$ such that $a\lt e\lt b$, then
\begin{equation}
\int_a^b\,f(x)dx=\int_a^e\,f(x)dx+\int_e^b\,f(x)dx.
\end{equation}

Answer: 

It is true that if $f(x)$ is continuous on $[a,b]$ and we have a point $e$ such that $a\lt e\lt b$, then \begin{equation} \int_a^b\,f(x)dx=\int_a^e\,f(x)dx+\int_e^b\,f(x)dx. \end{equation}