Show that the function defined by $f(x) = \sin(x^{2})$ is a continuous function.
(N/A) The function $f(x) = \sin(x^{2})$ is defined for all real numbers $x \in \mathbb{R}$.
We can express $f(x)$ as the composition of two functions $g(x)$ and $h(x)$,where $g(x) = \sin(x)$ and $h(x) = x^{2}$.
Then,$(g \circ h)(x) = g(h(x)) = g(x^{2}) = \sin(x^{2}) = f(x)$.
Since $g(x) = \sin(x)$ is a continuous function for all $x \in \mathbb{R}$ and $h(x) = x^{2}$ is a polynomial function which is continuous for all $x \in \mathbb{R}$,the composition of two continuous functions is also continuous.
Therefore,$f(x) = \sin(x^{2})$ is a continuous function for all $x \in \mathbb{R}$.
We can express $f(x)$ as the composition of two functions $g(x)$ and $h(x)$,where $g(x) = \sin(x)$ and $h(x) = x^{2}$.
Then,$(g \circ h)(x) = g(h(x)) = g(x^{2}) = \sin(x^{2}) = f(x)$.
Since $g(x) = \sin(x)$ is a continuous function for all $x \in \mathbb{R}$ and $h(x) = x^{2}$ is a polynomial function which is continuous for all $x \in \mathbb{R}$,the composition of two continuous functions is also continuous.
Therefore,$f(x) = \sin(x^{2})$ is a continuous function for all $x \in \mathbb{R}$.
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