Take a=0 and b=1, so that we're dealing with the unit interval. Look for a continuous function on the (closed) unit interval whose only fixed points are at the endpoints of the interval. If you think in terms of the graph of the function, you want a function f with f(0)=0 and f(1)=1, whose graph lies entirely under (or over) the diagonal of the unit square (except at the endpoints). That shouldn't be too hard to find.