Answer to Question #297049 in Real Analysis for John

Question

<e> Examine the f:R->R defined by

f(x)={1/6(x+1)^3 x is not equal to 0

5/6 x=0}

for continuity on R .If it is not continuous at any of R find the nature of discontinuity there

Answer 1

Since

$\lim\limits_{x\to 0}f(x)=\lim\limits_{x\to 0}\frac{1}6(x+1)^3=\frac{1}6\ne\frac{5}6=f(0),$

we conclude that $x=0$ is a point of removable discontinuity.

On the set $\R\setminus\{0\}$ the function $f$ is an elementary function, and hence it is continuous.