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 ${lim}_{x\to c}f\left(x\right)\ne f\left(c\right)$.

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

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

Let $g\left(x\right)=\{\begin{array}{cc}{x}^2+5& x<2\\ 10& x=2\\ 1+x^3& x>2\end{array}$ then $x=2$ :

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

Answer 1

Answer: (B)

is a removable discontinuity

$\underset{x\to 2+}{lim}g\left(x\right)=1+2^3=9$
$\underset{x\to 2-}{lim}g\left(x\right)=4+5=9$
So ${lim}_{x\to 2}g\left(x\right)$ exists but is not equal to $g\left(2\right)=10$
Thus at $x=2$ is removable discontinuity.