Discuss the continuity and differentiability of `f(x)={(x-c)cos(1/(x-c)),\ \ \ x!=c0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x=c`

Explanation

Given function,
`f(x)={{:((x-c)cos(1/(x-c)),x ne c),(0,x=c):}`
For the continuity at `x=c`

`f(c)=0`
`LHL=lim_(x->c^-)f(x)`
`=lim_(h->0)f(c-h)`
`=lim_(h->0)(c-h-c)cos(1/(c-h-c))`
`=lim_(h->0)(-h)cos(1/-h)`
`=0`
`RHL=lim_(x->c^+)f(x)`
`=lim_(h->0)f(c+h)`
`=lim_(h->0)(c+h-c)cos(1/(c+h-c))`
`=lim_(h->0)(h)cos(1/h)`
`=0`
since,`LHL=RHL=f(c)`
So, f(x) is continuous at `x=c`


For differentiablility at `x=0`
`LHD=lim_(x->c^-)(f(x)-f(c))/(x-c)`
`=lim_(h->0)(f(c-h)-f(c))/(c-h-c)`
`=lim_(h->0)((c-h-c)cos(1/(c-h-c))-0)/-h`
`=lim_(h->0)((-h)cos(1/-h))/-h`
`lim_(h->0)cos(1/h)=p`(let)
now,
`RHD=lim_(x->c^+)(f(x)-f(c))/(x-c)`
`=lim_(h->0)(f(c+h)-f(c))/(c+h-c)`
`=lim_(h->0)((c+h-c)cos(1/(c-h-c))-0)/-h`
`=lim_(h->0)((h)cos(1/h))/h`
`lim_(h->0)cos(1/h)=p`(let)
so,`LHD=RHD`
Hence, f(x) is continuous and differentiable at `x=c.`