Math Challenges

Solution:
\( \displaylines{\text{Let }g\left(x\right)=\left(x,f\left(x\right)\right).\\ \text{If }f\text{ is continuous, }g\text{ is continuous. Since }E\text{ is compact, the continuous image }g\left(E\right)\text{ is compact.}\\ \text{If we instead know that }g\left(E\right)\text{ is compact, this means that for a convergent sequence }x_{n}\to x\\ \text{we have}\left(x_{n},f\left(x_{n}\right)\right)\to\left(x,f\left(x\right)\right).\text{ Therefore }f\left(x_{n}\right)\to f\left(x\right)\text{ and }f\text{ is continuous.}} \)