I'm currently learning about the Laplace transform and in my textbook in college and I have this definiton:
Function $f$ is of exponential order if there exist constants $M>0$ and $a$ such that for every $t>0$ relation $|f(t)| \le Me^{at}$ holds.
I have the folloiwing assignment: let the function $f$ be continuous on $[0, \infty >$, prove that it is of exponential order when and only when there exist constants $a, M$ and a number $t_{0} > 0$ such that $|f(t)| \le Me^{at}$, for every $t > t_{0}$ holds.
How do I prove (go about proving) this? How do you usually go about proving something? Do proofs only need to contain math symbols or also some text? How to practice proving mathematical statements?
Best Answer
Hint: For any real $b$ we have $K(b)=\min \{e^{bx}: x\in [0,t_0]\}>0$, and since $f$ is continuous,we have $m(f)=\max \{|f(x)|:x\in [0,t_0]\}<\infty.$
So for all $x\in [0,t_0]$ and all $b\in \mathbb R$ we have $|f(x)|/e^{bx}\leq m(f)/K(b)<\infty.$
The idea is that if $|f(x)|\leq Me^{ax}$ for all $x>t_0> 0,$ we can find $M'\geq M$ and $b\geq |a|$ such that $|f(x)|\leq M'e^{bx}$ for all $x\geq 0.$