It's much, much stronger than continuous. If you're allowed to use the limit definition of continuity just note that

Note that by induction . Thus, we see thatb)Pick some point y1 in R and construct the sequence (y1,f(y1),f(f(y1)),....)

In general if y_(n+1)=f(yn) show that the resulting sequence yn is a Cauchy sequence. hence we may let y=limyn

From where it evidently follows, letting that is Cauchy, and thus by the completeness of it follows that exists.

Uniqueness is clear in general since if but then we see our conditions state thatc)Prove that y is a fixed point of f and that is unique in this regard.

Now, to see that is a fixed point we merely note that since that and so by the continuity of we have that

Merely note that was arbitrary to begin with.d) Finally prove that if x is any arbitrary point in R then the sequence (x,f(x),f(f(x)),...) converges to y defined in (b).

Remark:There is a much more elegant way of proving that a contractive map from a complete metric space to itself has a fixed point.