#### Sensing continuous Functions

When we examine the physical environment around us, we may want
to record our observations. Recording everything we observe is
rarely possible however and therefore we often try to single out
one elementary aspect and focus on this aspect alone. The most
basic type of such a recording is that of a single quantity that
we express as a real number \(r \in {\mathbb{R}}\). To obtain
such a value \(r\), we often use some sensor that we place
somewhere in our physical environment and then we read the
sensor’s value. In most situations the sensor will show a
different value, when we place it somewhere else and even when
we repeat the process and read the value from the sensor in the
same location another time, the value may be different. In the
latter situation the value is likely to be similar however and
in the former situation the value is likely to be similar, if
the new location is very close to the previous location. We
assume that there is a certain subspace \(X\) of the environment
around us within our reach and interest, where we can place the
sensor. As a subspace of our environment, \(X\) inherits a
topology
and the above observations lead to the intuition, that there is
a
proper
continuous
function \(f \colon X \rightarrow {\mathbb{R}}\) whose
values we are reading of the sensor in proximity. This is also
called a scalar field. Now we may never learn this function
\(f\), but we can interpolate between our samples from a
sufficiently dense set of measuring locations, to define another
continuous function \(g \colon X \rightarrow {\mathbb{R}}\) that
is close to \(f\) in the sense that
\(-\varepsilon \leq f(p) - g(p) \leq \varepsilon\) for all
\(p \in X\) and a real number \(\varepsilon > 0\) of moderate
size. More specifically we can make \(\varepsilon\) as small as
we like by increasing both, the density of our samples and the
accuracy of our sensor. We concede that the problem of
approximating \(f\) is non-trivial and that
interpolation
is a vast field outside the scope of this document.