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**lllll** Show that $\displaystyle f(x)=x^n$ where $\displaystyle n \in \mathbb{N}$ is continuous on the interval $\displaystyle [a,b]$.

using the $\displaystyle \delta \ \varepsilon $ yields:

$\displaystyle \forall \ \delta>0, \ \varepsilon >0 \ \ |x-c|<\delta, \ \mbox{then} \ |x^n-c^n|<\varepsilon$ for $\displaystyle n \in \mathbb{N}$ at which point I don't know how to continue.

I figure, I can write $\displaystyle f(x)= x\cdot x \cdot x \dotso \cdot x$. furthermore I can express this as a number of composite functions where I would have:

$\displaystyle f(x)=x^n=(f \circ f \circ f \circ \dotso \circ f)(x) $. This is wrong!

I know how to show that given two continuous function then their composite is continuous, but I don't know how I would show that n composite functions are continuous, I imagine this would have to be done by induction.