Chapter 10.1, Problem 1TFR

For Questions (a)-(i), decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem from the text. If false, provide an example, illustration, or brief explanation of why the statement is false.

(a) The function

$f\left(t\right)=\{\begin{array}{l}1,\text{if}t\text{is}\text{an}\text{integer,}\\ t^2,\text{if}t\text{is}\text{not}\text{an}\text{integer,}\end{array}$

is piecewise continuous on $\left[0,\infty \right)$.

(b) The function

$g\left(t\right)=\{\begin{array}{l}\frac{1}{\sin t},\text{if}t\text{is}\text{not}\text{a}\text{multiple}\text{of}\text{π},\\ 0,\text{if}t\text{is}\text{a}\text{multiple}\text{of}\text{π,}\end{array}$

is piecewise continuous on $\left[0,\infty \right)$.

(c) If $f$ and $g$ are piecewise continuous functions on an interval $I$, then so is $f+g$.

(d) If $f$ is piecewise continuous on the interval $\left[a,b\right]$ and on the interval $\left[b,c\right]$, then f is piecewise continuous on the interval $\left[a,c\right]$.

(e) The Laplace transform of a function $f$ is a function $F\left(s\right)$ defined by

$F\left(s\right)={\displaystyle {\int }_1^{\infty }{e}^{-st}f\left(t\right)}dt$,

provided that this improper integral converges.

(f) The Laplace transform $F\left(s\right)$ of $f\left(t\right)=e^t$ is only defined for $s>1$.

(g) The Laplace transform $F\left(s\right)$ of $f\left(t\right)=2\cos 3t$ is only defined for $s>3$.

(h) For any function $f$ such tat $f\left(x\right)>0$ for all $x$ in $\left[0,\infty \right)$, $L\left[\frac{1}{f}\right]=\frac{1}{L\left[f\right]}$.

(i) For any function $f$ such that $f\left(x\right)>0$ for all $x$ in $\left[0,\infty \right)$, $L\left[f^2\right]=L{\left[f\right]}^2$.