Determining if a Continuous Function is Differentiable | Steps and Criteria

to test if a continuous function is differentiable..

To test if a continuous function is differentiable, we need to check if it satisfies the differentiability criteria

To test if a continuous function is differentiable, we need to check if it satisfies the differentiability criteria. Here are the steps to determine if a continuous function is differentiable:

1. First, ensure that the function is continuous. A function is continuous if it does not have any jumps, holes, or vertical asymptotes. For example, a function like f(x) = 1 / (x – 2) is not continuous at x = 2.

2. Next, check if the function is differentiable within its domain. Differentiability implies that the function has a derivative at every point. The derivative represents the rate of change of the function at a specific point. If the derivative exists everywhere within the domain of the function, then the function is differentiable.

3. Calculate the derivative of the function. This can be done using differentiation rules such as the power rule, product rule, quotient rule, or chain rule. If the derivative can be computed at every point within the domain, then the function is differentiable.

4. Check for any singular points. A singular point is a point where the derivative does not exist or is not continuous. If there are any singular points within the domain of the function, then the function is not differentiable at those points.

5. Finally, check if the derivative is continuous. Even if a function has a derivative at every point, it must be continuous to be considered differentiable. If there are any discontinuities or jumps in the derivative, then the function is not differentiable.

It is important to note that differentiability is a stronger condition than continuity. A function can be continuous but not differentiable at certain points. For example, the function f(x) = |x| is continuous everywhere but not differentiable at x = 0 since the derivative has a sharp turn at that point.

In summary, to test if a continuous function is differentiable, you need to ensure that the function satisfies all the criteria mentioned above. If it passes all the tests, then the function is indeed differentiable.

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