I can't figure out how to prove the continuity of the following functions (I've tried searching to see if this question was posted already. I found someone post something about e^x, but no one replied and I also wanted the other functions, so sorry if this is a repeat of a topic already posted):

given some $\displaystyle c \in \mathbb{R}$

$\displaystyle \ln x, c^x, \sin x, \text{ and } cos x$

Given the following definition for continuity:

Let E be a nonempty subset of $\displaystyle \mathbb{R}$ f : E -> $\displaystyle \mathbb{R}$

f is continous at a point a $\displaystyle \in $ E iFF given $\displaystyle \epsilon > 0 \text{ } \exists\text{ } \delta > 0 \text{ } s.t. \text{ } |x-a|< \delta \Rightarrow \text{ } |f(x)-f(a)| < \epsilon$

I have cauchy, sequential characterization of limits, bolzano weirstrauss, squeeze theorem, limit comparison theorems, definition of bounded functions, limit of composite functions [if limit of f exists, and g is continuous, limit (g(f(x)) = g[lim (f(x)) ], extreme value theorem and intermediate value theorem at my disposal.

My intervals:

For log, I took my interval as (0, $\displaystyle \infty$)

For c^x ($\displaystyle - \infty$, $\displaystyle \infty$)

For sin/cos ($\displaystyle - \infty$, $\displaystyle \infty$)

For the ln x, I got down to:

$\displaystyle |x-a|< \delta$

$\displaystyle |ln (\frac{x}{a})|$ but I got stuck there. I thought about raising it to a power of e, but then I'd have to prove that e^x is continuous.

For the c^x, I got down to:

$\displaystyle |x-a|< \delta$

$\displaystyle |c^a||c^{x-a}-1|<|c^a||c^{\delta}-1|$ but I couldn't find it a way to make it less than epsilon.

For the sin x and cos x, I got down to:

$\displaystyle |x-a|< \delta$

$\displaystyle |sin x - sin a|$ and cos x, respectively, and I wasn't sure how to even simplify it to take advantage of $\displaystyle |x-a|< \delta$, much less show that $\displaystyle |\sin x - \sin a|< \epsilon$

I thought about using series expansion of sin/cos.