Differentiable means that a function has a derivative at a given point, while continuously differentiable means that the derivative of the function exists and is continuous over a given interval. In other words, a function is continuously differentiable if its derivative is a continuous function. So, all continuously differentiable functions are differentiable, but not all differentiable functions are continuously differentiable. **

The function is continuous and differentiable in the interval [n, 1/n] because it is a composition of continuous and differentiable functions. The function f(x) = 1/x is continuous and differentiable for all x ≠ 0, and the function g(x) = x is continuous and differentiable for all x. Since the composition of continuous functions is continuous, and the composition of differentiable functions is differentiable, the function h(x) = f(g(x)) = f(x) = 1/x is continuous and differentiable in the interval [n, 1/n]. **

Not necessarily. A function can be continuous without being differentiable. For example, the absolute value function is continuous everywhere but not differentiable at the point where the function changes direction. A function must satisfy certain conditions, such as having a well-defined tangent at each point, in order to be considered differentiable. **

No, 1z is not complex differentiable. In order for a function to be complex differentiable at a point, it must satisfy the Cauchy-Riemann equations, which require the function to be holomorphic. Since 1z is not holomorphic (as it does not satisfy the Cauchy-Riemann equations), it is not complex differentiable. **

No, not every antiderivative is continuously differentiable. While every antiderivative of a continuous function is continuous, it may not necessarily be continuously differentiable. For example, the antiderivative of the absolute value function, which is not continuously differentiable at the point where the function changes direction, is not continuously differentiable. Therefore, it is important to note that while antiderivatives are always continuous, they may not always be continuously differentiable. **

A function is continuous at a point if the limit of the function as it approaches that point exists and is equal to the value of the function at that point. A function is differentiable at a point if the derivative of the function exists at that point. A function is continuous and differentiable on an interval if it is continuous and differentiable at every point within that interval. **

In mathematics, a function is said to be differentiable at a point if it has a derivative at that point. This means that the function has a well-defined tangent line at that point, indicating how the function changes locally around that point. A function is differentiable on an interval if it is differentiable at every point within that interval. The concept of differentiability is fundamental in calculus and is used to study the rate at which functions change. **

The function is not differentiable at points where it has sharp corners, cusps, or vertical tangents. These points are called points of non-differentiability. Additionally, the function is not differentiable at points where it has discontinuities or breaks in its graph. At these points, the derivative of the function does not exist. **