If you define \cos(\theta) and \sin(\theta) in the usual geometric way, as the x- and y- coordinates of a point at angle \theta on the unit circle, and you're willing to accept a simple geometric argument that these coordinates must approach (1,0) as \theta approaches 0, then this means that \cos and \sin are continuous at 0. From that, you can use trigonometric identities to get continuity elsewhere, for example
\sin(x+h) - \sin(x) = \sin(x)\cos(h) + \cos(x)\sin(h) - \sin(x)
so
\begin{align*}<br />
\lim_{h \rightarrow 0} (\sin(x+h) - \sin(x)) &= \sin(x) \left(\lim_{h \rightarrow 0}\cos(h)\right) + \cos(x)\left(\lim_{h \rightarrow 0}\sin(h)\right) - \sin(x) \\<br />
&= \sin(x) \cos(0) + \cos(x) \sin(0) - \sin(x) \\<br />
&= \sin(x) + 0 - \sin(x) = 0<br />
\end{align*}
and therefore \sin is continuous at x. I'm not sure how you would turn this into a fully rigorous \epsilon-\delta argument, however.
