What is the definition of differentiability? Some places define it as:

If the Left hand derivative and the Right hand derivative at a point are equal then the function is said to be differentiable at that point.

Others define it based on the condition of the existence of a unique tangent at that point.
Which one of these is correct? Or are both of them wrong? 
According to the first definition, the curve need not be continuous at that point and can have a point discontinuity or a hole, like this:

Moreover it doesn't stop a curve, with a jump discontinuity but with same slope on both sides of it, from being differentiable.
And lastly, if a function is not defined at a point, then is the function discontinuous there too?

A function is differentiable (has a derivative) at point x if the following limit exists: 
$$
\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}
$$
The first definition is equivalent to this one (because for this limit to exist, the two limits from left and right should exist and should be equal). But I would say stick to this definition for now as it's simpler for beginners. The second definition is not rigorous, it is quite sloppy to say the least.  
Also, there's a theorem stating that: if a function is differentiable at a point x, then it's also continuous at the point x.  
