7.4. Local AspectsAs with continuity differentiability at a proves to be local property. It is e.g. compatible with restricting functions to a subset of their domain:
So we have for instance for any subset A of $\mathbb{R}$ with an accumulation point $a\in A$:
$(cA{)}^{\prime}(a)=0$
and $(XA{)}^{\prime}(a)=1$
Two functions that coincide locally have the same differentiability behaviour, a feature that also reflects the local character.
Consider:
Functions defined by sections could be treated niftily using [7.4.2]. We take the absolute value function to demonstrate this.
According to the above mentioned example in 7.3 the absolute value function is not differentiable at 0. [7.4.3] thus proves that it fails to be differentiable at only one point. The next example introduces a function being the exact opposite, i.e. a function which is solely differentiable at a single point. Both examples show that the differential quality of a function at one point has no impact on that quality with the neighbouring points.
Another local aspect is the problem to concatenate two functions (cf [6.8.7]), in this case to concatenate them differentiably. The following proposition shows that this is merely the question whether two functions coincide in their values and their derivative numbers at a.
