Is the following function $f(x)$ continuous? I have a problem with this question: 
How can I prove that the following function is continuous, if it is?
$$f(x)=\frac{ \left( \frac{\cos(\cos(\cos \left(\cos(\cos(x))))\right)}{\sin\left(\sin\left(\sin\left(e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^x}}}}}}}}}}}\right)\right)\right)} \right) }{\ln\left(2x+1\right)+2+\cos x}$$
 A: This function is not continuous at the following points (as it is not defined): When $\ln(2x+1)+2+\cos(x)=0$ (which occurs at $\approx x=-.47223$), and whenever $e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^x}}}}}}}}}}}=k\pi$ for $k$ a positive integer. As well, $\ln(2x+1)$ is undefined for $x\leq-.5$, so the function in question is as well. Otherwise, the function is defined, and is a composition of continuous functions, and hence is continuous.
A: No it isn't. Observe that $$\lim_{x \to -\frac 12^+} \ln (2x+1)+2 +\cos x = - \infty$$
and also
$$\ln (2x+1)+2 +\cos x \ \bigg |_{x = \frac \pi 2} = \ln (\pi +1) +2 >0$$
So, by Intermediate Value Theorem, there is a point $x_0 \in (- \frac 12, \frac \pi 2 )$ such that $$\ln (2x+1)+2 +\cos x \ \bigg |_{x = x_0} = 0$$
Thus, the denominator is zero at $x_0$ and therefore $f$ is discontinuous.
A: This function is continuous on its domain.  (Just what it's domain is, is a more delicate question.)


*

*The cosine, sine, logarithm, constant, and exponential functions are continuous.

*Sums, quotients, and products of continuous functions are continuous.

*Compositions of continuous functions are continuous.


All three bullet points above require proof.  All three are standard results.  Once established, these three are enough to deal with this proposed function.
