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A Quick Look at Functions
The Topology of the Real Numbers
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absolute value accumulation point attracting fixed point attracting periodic point bifurcation diagram Cantor set chaotic Chapter closed intervals codomain complex numbers complex plane complex quadratic polynomial contained continuous function converges defined Definition demonstrated dense differential equation digits dynamical systems element Euler approximation eventually fixed eventually periodic Example Exercise Set exists filled Julia set function f graph graphical analysis Hence homeomorphism hyperbolic integer Intermediate Value Theorem intersection irrational number iteration Mandelbrot set mathematical metric space natural number neighborhood Newton's function Newton's method Note one-to-one open interval open set parameter values period doubling bifurcation periodic orbit point of f point with prime prime period Proposition 3.13 qc(z qualitative behavior radius rational numbers reader real numbers repelling fixed point root rx(l Sarkovskii's Theorem sequence set of infinity shown in Figure solution stable set subset Suppose Theorem 4.2 topological conjugacy topologically conjugate topologically transitive Triangle Inequality