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What is the difference between differentiable and continuously differentiable?
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.

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

Is a continuous function differentiable?
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 welldefined tangent at each point, in order to be considered differentiable.

Is every antiderivative continuously 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.

What is the definition of differentiable?
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 welldefined 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.

Are these graphs continuous and differentiable?
Yes, both graphs are continuous as there are no breaks or jumps in the lines. However, the first graph is not differentiable at the point where the line changes direction abruptly, as there is a sharp corner. The second graph is differentiable everywhere as it has a smooth curve without any sharp corners or cusps.

Where is the function not differentiable?
The function is not differentiable at points where it has sharp corners, cusps, or vertical tangents. These points are called points of nondifferentiability. 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.

Are all continuous monotonic functions differentiable?
No, not all continuous monotonic functions are differentiable. While all differentiable functions are continuous and monotonic, the reverse is not necessarily true. For example, the absolute value function is continuous and monotonic, but it is not differentiable at the point where the function changes direction. Therefore, it is important to note that while continuous monotonic functions often are differentiable, it is not a guarantee.

Why are broken rational functions differentiable?
Broken rational functions are differentiable because they can be expressed as a quotient of two differentiable functions. The quotient rule for differentiation allows us to differentiate the numerator and denominator separately and then combine the results using the formula for the derivative of a quotient. As long as the denominator is not zero, the broken rational function will be differentiable. This is because the derivative of a quotient of differentiable functions is defined for all values of x where the denominator is not zero.

At which points is this function differentiable?
The function is differentiable at all points except at the points where the absolute value function changes direction abruptly, such as at x = 0. At these points, the function is not smooth and has a sharp corner, making it nondifferentiable. Everywhere else, the function is continuous and smooth, allowing for differentiability.

Why is fx1x differentiable, but not continuous?
The function f(x) = x is not differentiable at x = 0 because its derivative does not exist at that point. However, it is continuous everywhere. On the other hand, the function g(x) = x^2 is differentiable everywhere, including at x = 0, because its derivative exists at every point. This is because the function x^2 is a composition of the functions x and x^2, both of which are differentiable everywhere. Therefore, the function f(x) = x is not differentiable at x = 0, but g(x) = x^2 is differentiable everywhere.

In which points is the function differentiable?
The function is differentiable at all points where it is continuous and has a welldefined tangent line. This includes points where the function has a smooth and continuous curve without any sharp corners or cusps. Additionally, the function is differentiable at points where it has a sharp corner or cusp, as long as the derivative from the left and right sides match. However, the function is not differentiable at points where it has a vertical tangent line or a discontinuity.