A function is considered differentiable if its derivative exists at each point in its domain. In other words, a function is differentiable everywhere its derivative is defined. To determine differentiability, we can use limits and continuity. The definition of differentiability is expressed as follows:

• A function f is differentiable on an open interval (a,b) if the limit of (\lim _{h \rightarrow 0} \frac{f(c+h)-f(c)}{h}) exists for every c in (a,b) .
• If f is differentiable, meaning (f^{\prime}(c)) exists, then f is continuous at c.
• For a function to be differentiable, it must be continuous, and its derivative must be continuous as well.

Therefore, continuity is a necessary condition for differentiability. However, not all continuous functions are differentiable. A function can fail to be differentiable at a point if:

• The function is not continuous at the point.
• The graph has a sharp corner at the point.
• The graph has a vertical line at the point.

In summary, a function is differentiable if its derivative exists at each point in its domain. To determine differentiability, we can use limits and continuity. Continuity is a necessary condition for differentiability, but not all continuous functions are differentiable. A function can fail to be differentiable at a point if it is not continuous at that point, has a sharp corner, or has a vertical line.