Differentiability of a function at a point is determined by checking if the derivative of the function exists at that point. If the derivative exists, then the function is said to be differentiable at that point. The derivative can be found using various techniques such as the limit definition of the derivative, rules of differentiation, or by checking if the function is continuous at that point. If the derivative exists, the function is considered differentiable at that point; if not, it is not differentiable. **

Differentiability continuously means that a function is not only differentiable at a specific point, but its derivative is also continuous at that point. This implies that the function has a well-defined tangent line at that point, and that the rate of change of the function is consistent in the neighborhood of that point. In other words, the function's derivative does not have any sudden jumps or breaks at that point, and it smoothly transitions from one value to another. **

One can prove differentiability of a function at a point by showing that the limit of the difference quotient exists as the independent variable approaches that point. This can be done by calculating the derivative of the function at that point using the definition of the derivative, or by using known rules and properties of differentiable functions. Additionally, one can also use the concept of continuity to show that a function is differentiable at a point, as differentiability implies continuity. **

Differentiability implies continuity because in order for a function to be differentiable at a point, it must be continuous at that point. This is because the definition of differentiability includes the existence of a derivative, which in turn requires the function to be continuous. If a function is not continuous at a point, it cannot have a derivative at that point, and therefore cannot be differentiable. Therefore, differentiability implies continuity. **

To show continuity of a function at a point, we need to demonstrate that the limit of the function as it approaches that point exists and is equal to the value of the function at that point. This can be done using the epsilon-delta definition of continuity. To show differentiability at a point, we need to demonstrate that the derivative of the function exists at that point, which can be done by showing that the limit of the difference quotient exists as it approaches that point. Additionally, we can use the definition of differentiability to show that the function is continuous at that point. **

To show total differentiability at a point, you need to demonstrate that all partial derivatives exist and are continuous at that point. This means calculating the partial derivatives with respect to each variable and ensuring they are continuous functions. Additionally, you can use the definition of total differentiability, which states that a function is totally differentiable at a point if it can be approximated by a linear transformation at that point. This can be shown by proving that the function can be written as the sum of its linearization and a remainder term that approaches zero as the point approaches the given point. **

Continuous differentiability refers to a function that has a derivative at every point in its domain, and that derivative function is itself continuous. In other words, a function is continuously differentiable if it is differentiable at every point and the derivative function is also continuous. This means that the rate of change of the function is smooth and continuous throughout its domain. Functions that are continuously differentiable are often used in mathematical modeling and optimization problems. **

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