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# Show that the function defined by f (x) is equal to cos (x raised to 2) is a continuous function.

31. Show that the function defined by$f (x) = \cos (x^2 )$ is a continuous function.

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Given function is
$f (x) = \cos (x^2 )$
given function is defined for all real values of x
Let x = k + h
if $x\rightarrow k , \ then \ h \rightarrow 0$
$f(k) = \cos k^2\\ \lim_{x \rightarrow k}f(x) = \lim_{x \rightarrow k}\cos x^2 = \lim_{h \rightarrow 0}\cos (k+h)^2 = \cos k^2\\ \lim_{x \rightarrow k}f(x) = f(k)$
Hence, the function  $f (x) = \cos (x^2 )$ is a continuous function

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