Use the definition of continuity to prove that the constant function \(g(x)=c\) is continuous at any point \(a\text{.}\)
Problem8.5.2.
(a)
Use the definition of continuity to prove that \(\ln x\) is continuous at \(1\text{.}\)
Hint.
You may want to use the fact \(\abs{\ln x}\lt \eps\,\Leftrightarrow-\eps\lt \ln x\lt \eps\) to find a \(\delta\text{.}\)
(b)
Use part (a) to prove that \(\ln x\) is continuous at any positive real number \(a\text{.}\)
Hint.
\(\ln(x)=\ln(x/a)+\ln(a)\text{.}\) This is a combination of functions which are continuous at \(a\text{.}\) Be sure to explain how you know that \(\ln(x/a)\) is continuous at \(a\text{.}\)
Problem8.5.3.
Write a formal definition of the statement \(f\) is not continuous at \(a\text{,}\) and use it to prove that the function \(f(x)= \begin{cases}x\amp \text{ if } x\neq 1\\ 0\amp \text{ if } x=1 \end{cases}\) is not continuous at \(a=1\text{.}\)
