Definition: We say that a function $f$ is continuous at a provided that for any $\epsilon > 0$, there exists a $\delta > 0$ such that if $|x−a| < \delta$ then $|f(x)−f(a)| < \epsilon$.
(a) Use the definition of continuity to prove that $\ln(x)$ is continuous at $1$. [Hint: You may want to use the fact $|\ln(x)| < \epsilon \iff − \epsilon < \ln(x) < \epsilon$ to find a $\delta$.]
(b) 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 that are continuous at $a$. Be sure to explain how you know that $\ln(x/a)$ is continuous at $a$.]
For part (a) how can I find $\delta$ that works for if $|x−a| < \delta$ then $|f(x)−f(a)| < \epsilon$, and for part (b) how can I show that $\ln(x/a)$ is continuous at $a$?
Please help me with parts (a) and (b).