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 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.
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.