Differentiability and continuity

Hace a problem that asks me to determine if the following function;

f(x) = if x<0

and , if

is continuous at x = 0 and then if it's differentiable at x=0,

I figured to compute the limit as x -> 0 from x<0 and x>0 to determine continuity.

when i did lim(x->0) i used lhopitals to give 1/1 which isnt equal to f(0) = 0 so i figured the function wasn't continuous.

But i assume i have made a mistake here as when i went ahead and checked for differentiability i found the function to be diff at x=0. which doesn't make sense as if differentiable it must be continuous?

Please help.

Cheers.