Differentiation—Fundamentals
In the previous chapter, we studied continuous functions. Geometrically, a real functions is continuous on a interval if the graph is unbroken. Here, we consider differentiable functions. These are the continuous functions whose graphs are smooth, i.e. have a tangent at each point. The reader should note that not every curve is smooth. A circle is smooth since it has a tangent at each point (the tangent being perpendicular to the radius at the point). On the other hand, a square is not smooth since there is no tangent at any vertex.
The first section establishes an analytic characterisation of (the gradient of) the tangent to a graph in terms of a limit. Our knowledge of limits allows is to decide whether ...
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