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

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  • If $f$ is a real function on a subset of the real numbers and $c$ be a point in the domain of $f$, then $f$ is continuous at $c$ if $\lim\limits_{\large x\to c} f(x) = f(c)$.
Step 1:
$f(x)=\cos x^2$
Let $g(x)=\cos x$
$g(h).(x)=g(h(x))=\cos x^2$
Step 2:
Now $g$ and $h$ both are continuous for all $x\in R$
$f(x)=(goh)(x)=\cos x^2$ is also continuous at all $x\in R$
answered May 29, 2013 by sreemathi.v

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