Show that the function defined by f(x) = (x^2) is a continuous fu… - Vidyadip

Question

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

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Answer

Observe that the function is defined for every real number. The function $f$ may be thought of as a composition g o h of the two functions g and h , where $\mathrm{g}(\mathrm{x})=\sin \mathrm{x}$ and $\mathrm{h}(\mathrm{x})=\mathrm{x}^2$. Since both g and h are continuous functions, Suppose $f$ and $g$ are real valued functions such that $(f \circ g)$ is defined at $c$. If $g$ is continuous at $c$ and if $f$ is continuous at $\mathrm{g}(\mathrm{c})$, then $(\mathrm{f} \circ \mathrm{g})$ is continuous at c .

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