# Thread: continuity of basic functions

1. ## continuity of basic functions

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 $c \in \mathbb{R}$

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

Given the following definition for continuity:

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

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

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, $\infty$)

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

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

For the ln x, I got down to:

$|x-a|< \delta$

$|ln (\frac{x}{a})|$ 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:

$|x-a|< \delta$

$|c^a||c^{x-a}-1|<|c^a||c^{\delta}-1|$ 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:

$|x-a|< \delta$

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

I thought about using series expansion of sin/cos.

2. Were you given this a homework? Normally, these functions are defined in such a way as to make them continuous. What are your definitions of these functions?

As far as n! is concerned, since n! is only defined for n a non-negative integer, you can't take the limit "as n approaches a" and continuity doesn't really makes sense for such a function.

3. oh hmm, lol that's true huh. n! goes from N to N.

Yeah, it's part of an assignment, and I wasn't sure how to go about it. Though the ln x, wasn't I just figured, if I could prove either logarithms were continuous then I could use it to prove that exponential functions are continuous.

Though that means I could use the series expansions of a^x, and sin/cos as long as I can show that laurent polynomials are continuous.

4. You should really answer Hall's question: "What are your definitions of these functions?"