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

Let f(x) $=$ cos$({x^2})$. Domain of f $=$ R.

Let a be any arbitrary real number.

Then,$\mathop {\lim }\limits_{x \to {a^ + }} f(x) = \mathop {\lim }\limits_{\scriptstyle x \to a + h\atop\scriptstyle h \to 0} \cos {(a + h)^2} = \cos {a^2}$

Then, $\mathop {\lim }\limits_{x \to {a^ - }} f(x) = \mathop {\lim }\limits_{\scriptstyle x \to a - h\atop\scriptstyle h \to 0} \cos {(a - h)^2} = \cos {a^2}$ and $f(a) = \cos {a^2}$

Thus, $\mathop {\lim }\limits_{x \to {a^ - }} f(x) = \mathop {\lim }\limits_{x \to {a^ + }} f(x) = f(a)\forall a \in R.$

therefore, $f(x) = \cos ({x^2})$ is continuous at $a\forall a \in R.$