Show that the function defined by f (x) = cos (x2) is a continuous function.


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


The given function is $f(x)=\cos \left(x^{2}\right)$

This function f is defined for every real number and f can be written as the composition of two functions as,

$f=g \circ h$, where $g(x)=\cos x$ and $h(x)=x^{2}$

$\left[\because(g o h)(x)=g(h(x))=g\left(x^{2}\right)=\cos \left(x^{2}\right)=f(x)\right]$

It has to be first proved that (x) = cos x and h (x) = x2 are continuous functions.

It is evident that g is defined for every real number.

Let c be a real number.

Then, g (c) = cos c

Put $x=c+h$

If $x \rightarrow c$, then $h \rightarrow 0$

$\begin{aligned} \lim _{x \rightarrow c} g(x) &=\lim _{x \rightarrow c} \cos x \\ &=\lim _{h \rightarrow 0} \cos (c+h) \\ &=\lim _{h \rightarrow 0}[\cos c \cos h-\sin c \sin h] \\ &=\lim _{h \rightarrow 0} \cos c \cos h-\lim _{h \rightarrow 0} \sin c \sin h \\ &=\cos c \cos 0-\sin c \sin 0 \\ &=\cos c \times 1-\sin c \times 0 \\ &=\cos c \end{aligned}$

$\therefore \lim _{x \rightarrow c} g(x)=g(c)$

Thereforeg (x) = cos x is continuous function.


Clearly, h is defined for every real number.

Let $k$ be a real number, then $h(k)=k^{2}$

$\lim _{x \rightarrow k} h(x)=\lim _{x \rightarrow k} x^{2}=k^{2}$

$\therefore \lim _{x \rightarrow k} h(x)=h(k)$

Thereforeh is a continuous function.

It is known that for real valued functions and h,such that (h) is defined at c, if is continuous at and if is continuous at (c), then (g) is continuous at c.

Therefore, $f(x)=(g o h)(x)=\cos \left(x^{2}\right)$ is a continuous function.

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