Contrasts with the theory of functions of a real variable, (a) Consider f(x) = s

Question

Contrasts with the theory of functions of a real variable, (a) Consider f(x) = sin 1/x. Show that f is differentiable for all x notequalto 0 and bounded for all x. (b) Show that there is no differentiable function g(x) such that f(x) = g(x) for all x notequalto 0. Which aspect of the theory of analytic functions did we contrast with this example? (c) Define (x) = x^2 sin 1/x, x notequalto 0, (0) = 0. Show that is differentiable for all x. Is the zero of isolated at x = 0? Which aspect of the theory of analytic functions did we contrast? (d) Define phi(x) = e-^1/x^2, x notequalto 0, phi (0) = 0. Show that phi has derivatives of all order Series and Laurent Series at x = 0. Explain why phi (x) cannot possibly have a Maclaurin series representation. Which aspect of the theory of analytic functions did we contrast?

Explanation / Answer

a> we have f(x)= sin(1/x)

now let g(x) = sin(x) and h(x) = 1/x and both g and h are defined for x not equal to 0

THis tell us that f(x) is a composite function of g(x) and h(x)

and if g and h are differentiable for all x excluding 0 then f( x) will be differentiable for all x except x=0.

now g(x) = sinx is differentiable for the given domain x E R - {0}

and h(x) = 1/x is defined for all x E R - {0}

hence f(x) = g(h(x)) = sin(1/x) is differentiable for all x E R - {0}

f'(x) = cos(1/x)*(-1/x^2) = -1/x^2*cos(1/x) , for all x E R - {0}

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