Question

The continuity on an interval has a geometric interpretation. namely, a function f defined on an interval I is continuous on I if its graph has no 'holes' or 'jumps' .f is said to have a removable discontinuity at c if f(x) has a limit at c but lim $\underset{x\to c}{lim}f\left(x\right)\ne f\left(c\right)$.

If $\underset{x\to c+}{lim}f\left(x\right)and\underset{x\to c-}{lim}f\left(x\right)$ exist but are not equal then c is called jump discontinuity.

If $\underset{x\to c+}{lim}f\left(x\right)and\underset{x\to c-}{lim}f\left(x\right)$ fail to exist then c is called infinite discontinuity.

For $f\left(x\right)=\sin x$ , $x=0$ :

• A. is a point of continuity
• B. is a jump discontinuity
• C. is a removable discontinuity
• D. is an infinite discontinuity

Answer 1

The correct option is A is a point of continuity

$\underset{x\to 0}{lim}f\left(x\right)=\underset{x\to 0}{lim}\sin x=\sin 0=0$

$f\left(0\right)=0$

$\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{x\to 0^{+}}{lim}\sin x=\sin 0=0$

$\underset{x\to 0^{-}}{lim}f\left(x\right)=\underset{x\to 0^{-}}{lim}\sin x=0$

$\therefore$Point of continuity