Abstract
In writing this book my assumption has been that you have encountered the fundamental ideas of analysis (function, limit, continuity, differentiation, integration) in a standard course on calculus. For many purposes there is no harm at all in an informal approach, in which a continuous function is one whose graph has no jumps and a differentiable function is one whose graph has no sharp corners, and in which it is “obvious’ that (say) the sum of two or more continuous functions is continuous. On the other hand, it is not obvious from the graph that the function f defined by
is continuous but not differentiable at x=0, since the function takes the value 0 infinitely often in any interval containing 0, and so it is not really possible to draw the graph properly.
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References
Joseph Fourier, 1768–1830
Godfrey Harold Hardy 1877–1947
George Boole, 1815–1864
René Descartes, 1596–1650
Archimedes of Syracuse, 287–212 BC
Leopold Kronecker, 1823–1891
Blaise Pascal, 1623–1662
Augustin-Louis Cauchy, 1789–1857
Karl Hermann Amandus Schwarz, 1843–1921
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© 2001 Springer-Verlag London
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Howie, J.M. (2001). Introductory Ideas. In: Real Analysis. Springer Undergraduate Mathematics Series. Springer, London. https://doi.org/10.1007/978-1-4471-0341-7_1
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DOI: https://doi.org/10.1007/978-1-4471-0341-7_1
Publisher Name: Springer, London
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