Question

If $f\left(x\right)=\{\begin{array}{c}1,when0<x\le \frac{3\pi }{4}\\ 2sin\frac{2}{9}x,when\frac{3\pi }{4}<x<\pi \end{array}$ , then

• A. f(x) is continuous at x = 0
• B. f(x) is continuous at x = $\pi$
• C. f(x) is continuous at $x=\frac{3\pi }{4}$
• D. f(x) is discontinuous at $x=\frac{3\pi }{4}$

Answer 1

The correct option is C f(x) is continuous at $x=\frac{3\pi }{4}$

Here $f\left(\frac{3\pi }{4}\right)=1and{lim}_{x\to \frac{3\pi }{4-}}f\left(x\right)=1$
${lim}_{x\to \frac{3\pi }{4+}}f\left(x\right)={lim}_{h\to 0}2sin\frac{2}{9}(\frac{3\pi }{4}+h)=2sin\frac{\pi }{6}=1$
Hence f(x) is continuous at $x=\frac{3\pi }{4}$