When is derivative continuous

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The unofficial elections nomination post. Linked 0. Related Hot Network Questions. Question feed. Thus, is not a continuous function at 0. For details, see square times sine of reciprocal function First derivative. The basic idea for the example comes from the observation that derivative of differentiable function satisfies intermediate value property. This means that for any counterexample, the derivative would still satisfy the intermediate value property.

The next step is to therefore look for examples showing that intermediate value property not implies continuous. We see that the example used there is.

Unfortunately we don't have an elementary expression for the antiderivative of the sine of reciprocal function. However, we don't quite need to get an antiderivative of that function -- we just need something whose derivative is a sum of something that clearly goes to zero, and something of the structure of for instance, will also do.

The first attempt might be , since integration gets you one degree higher. Hence, differentiability is when the slope of the tangent line equals the limit of the function at a given point. This directly suggests that for a function to be differentiable, it must be continuous, and its derivative must be continuous as well. Now, this leads us to some very important implications — all differentiable functions must therefore be continuous, but not all continuous functions are differentiable!

Simply put, differentiable means the derivative exists at every point in its domain. Consequently, the only way for the derivative to exist is if the function also exists i. Thus, a differentiable function is also a continuous function.

We can easily observe that the absolute value graph is continuous as we can draw the graph without picking up your pencil.