How to prove these 6 functions to be continuous?

Question

Functions:

$ f(x) = c \\ f(x) =x \\ f(x) = e^x \\ f(x) = ln(x) \\ f(x) = sin(x)\\f(x) = \lvert x \rvert$

Please use the delta-epsilon definition of continuity to prove these.

Edit:

My proof for f(x) = c

Theorem: limit of x-> 0 of c

$Want\;to\;show: \forall \epsilon>0,\exists\delta>0,s.t.\lvert x \rvert < \delta \Rightarrow \lvert c-c\rvert<\epsilon \\ Let \;\epsilon>0\\Take\;\delta=\cfrac{\epsilon}{10}\\Assume\;\lvert x \rvert < \delta\\\mathrm{Therefore, \;since\;0\leq \lvert x \rvert < \delta\;,\lvert c-c\rvert\leq\lvert x \rvert < \delta\leq\epsilon}$

Answer 1

That proof went very well. Note that $\delta=\epsilon$ works. The "since" part is redundant. Now you should try with $f(x)=x$. Which would be the $\delta$?