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

Given the following definition for continuity:

Let E be a nonempty subset of f : E ->

f is continous at a point a E iFF given

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, )

For c^x ( , )

For sin/cos ( , )

For the ln x, I got down to:

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:

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:

and cos x, respectively, and I wasn't sure how to even simplify it to take advantage of , much less show that

I thought about using series expansion of sin/cos.