Question:
If $f(x)=x \sin \left(\frac{\pi}{4}\right)$ is everywhere continuous, then $f(0)=$______
Solution:
$f(x)=x \sin \left(\frac{\pi}{4}\right)=x \times \frac{1}{\sqrt{2}}=\frac{x}{\sqrt{2}}$
Thus, f(x) is a polynomial function which is continuous everywhere.
So, $f(x)$ is continuous at $x=0$.
$\therefore f(0)=\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0} \frac{x}{\sqrt{2}}=\frac{1}{\sqrt{2}} \times 0=0$
If $f(x)=x \sin \left(\frac{\pi}{4}\right)$ is everywhere continuous, then $f(0)=$ ____0____.
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