6.E: Continuity - What It Isn’t and What It Is (Exercises)

Q1

Use the definition of continuity to prove that the constant function \(g(x) = c\) is continuous at any point a.

Q2

1. Use the definition of continuity to prove that \(\ln x\) is continuous at \(1\). [Hint: You may want to use the fact \(\left |\ln x \right | < \varepsilon \Leftrightarrow -\varepsilon < \ln x < \varepsilon\) to find a \(δ\).]
2. Use part (a) to prove that \(\ln x\) is continuous at any positive real number \(a\). [Hint: \(\ln (x) = \ln (x/a) + \ln (a)\). This is a combination of functions which are continuous at \(a\). Be sure to explain how you know that \(\ln (x/a)\) is continuous at \(a\).]

Q3

Write a formal definition of the statement \(f\) is not continuous at \(a\), and use it to prove that the function \(f(x) = \begin{cases} x & \text{ if } x\neq 1 \\ 0 & \text{ if } x= 1 \end{cases}\) is not continuous at \(a = 1\).