Let us consider f(x)-f(0)/x-0=e^1/x/1+e^1/x=1-1/1+e^1/x.

The main thing we shud know is e^1/x tends to infinity as xtends to 0+ and it tends to 0 as x tends to 0-.So,1-1/1+e^1/x tends to 1 as x tends to 0+ and it tends to 0 as x tends to 0-.

Thus,Both right handed and left handed derivatives exist but are not equal.So,the fun. is continuous at x=0 but not differentiable.