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Continuity and Differentiability: Relation of Function Continuity to Function Differentiable at a Point

In The Real Numbers and Real Analysis (Bloch, 2011) the idea of a continuous function is stated as being somewhat intuitive. It is a function with no gaps or jumps. If there is a gap or jump in the function it is said to be discontinuous. In order to study whether a function is continuous or discontinuous, we need to study limits, because there is a close relationship between continuity and limits see Lemma 3.3.2. (Bloch, 2011) also reports that a derivative is intuitively used to resolve the issue of the rate of change of a function. By Definition 4.2.1 (Bloch, 2011) it can be seen that a function being differentiable is also closely associated with limits, and for this reason we also need to study limits.

In order to write of a proof to explain the connection between continuity of a function at a point and the function being differentiable at a point it will be necessary to study chapters three and four of the Bloch text. Does the connection between differentiable and continuity have to do with the reliance of both on limits? At a minimum the scratch for the proof will have investigate the definitions of both continuity and differentiable. Theorem 4.24 and Definitions 4.26 through 4.28 are related to continuous functions and functions being differentiable. One way the proof could take place is prove these various definitions directly or create a contradiction to prove the theorem or definitions. If we can show a function is differentiable at a point then it must be Relation of Function Continuity to Function Differentiable at Point 2 continuous at the same point, and then show the converse is not true this should be proof of the connection between continuity and differentiable. This topic is appropriate for a synchronous live oral defense because I believe that the level of detail can be delivered in a ten to fifteen minute time span, with a clear logical argument.

References
Bloch, E. D. (2011). The real numbers and real analysis. New York: Springer.

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Continuity and Differentiability

Let f : I → R be a function.
We start by defining the notion of continuity at a point a. Then we talk about the differentiability at a. We show the equivalence between the two definitions of continuity when h → 0 and when x → a. Next, we use the fact that a continuous function at a point a is bounded in a neighbourhood of a. That fact will allow us to prove that a differentiable function is also continuous. We conclude by showing the reverse is not true, a continuous function at the point a is not always differentiable at a.

Definitions and Properties
R is the set of real numbers. All the functions considered will be defined on an open interval I of R, meaning that I is of the form (a, b) where a is a real number or a = −∞ and b is either a real number strictly greater than a or b is ∞....
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