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Section 8.5 Additional Problems
Problem 8.5.1 .
Use the definition of continuity to prove that the constant function
\(g(x)=c\) is continuous at any point
\(a\text{.}\)
Problem 8.5.2 .
(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
\(a\text{.}\)
Problem 8.5.3 .
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
\(a=1\text{.}\)
