Show that the function `f` defined as follows `f(x)={3x-2,\ \ \ \ 02` is continuous at `x=2` , but not differentiable thereat.

Explanation

Given function,
`f(x)={{:(3x-2,0ltxle1),(2x^2-x,1lt x le2),(5x-4,xgt2):}`
LHL of f(x) at `x=2` is `2(2) ^2 −2=6`

RHL of f(x) at `x=2` is `5(2)−4=6`


So, f(x) is continuous at `x=2`.


Now let's check its derivability at `x=2`,

LHD of f(x) at `x=2` ,
`lim_(h->0)(f(2-h)-f(2))/(2-h-2)`
`=(2(2-h)^2-(2-h)-2(2)^2+2)/-h`
`=(8+2h^2-8h-2+h-6)/-h=7`
RHD of f(x) at `x=2`,
`lim_(h->0)(5(2+h)-4-5(2)+4)/(2+h-2)=5`
So, f(x) is not differentiable a `x=2.`