Let `f(x) = {{:(x exp[((1)/(|x|)+(1)/(x))]",",x ne 0),(0",",x = 0):}` Test whether (a) f(x) is continuous at x = 0
(b) f(x) is differentiable at x = 0

Options

A. discontinuous everywhere
B. continuous as well as differential for all x
C. continuous for all c but not differential at x=0
D. neither differential nor continuous at x=0

Answer

C

Explanation

We have
`f(x)={{:(,xe^-(-(1)/(x)+(1)/(x))=x,x lt 0),(,xe^(-2//x),x gt 0),(,0,x=0):}`
`"Clearly", underset(x to 0^(-))lim f(x)=underset(x to 0)lim x=0`
`underset(x to 0^(+))lim f(x)=underset(x to 0)lim xe^(-2//x)=0xx0=0 and, f(0)=0`
`therefore underset(x to 0^(-))lim f(x)=underset(x to 0^(+))lim f(x)=f(0)`
So, f(x) is continuous at x=0
`"Also", ("LHD at x=0")=((d)/(dx)(x))_("at x=0")=1`
`("RHD at x=0")={2^(-2//x)+(2e^(-2//x))/(x)}_("at x=0")`, which does not exist
Thus, f(x) is everywhere continuous but not differerntiabl at x=0