10. Continuity and Limits of Functions

If we want to compute the value of a specific function at some point a, it may happen that we can compute only the values of the function near a. Consider, for example, the distance a free-falling object covers. This is given by the equation \(s(t) = g \cdot t^{2}/2\), where t is the time elapsed, and g is the gravitational constant. Knowing this equation, we can easily compute the value of s(t). If, however, we want to calculate s(t) at a particular time t = a by measuring the time, then we will not be able to calculate the precise distance corresponding to this given time; we will obtain only a better or worse approximation—depending on the precision of our instruments. However, if we are careful, we will hope that if we use the value of t that we get from the measurement to recover s(t), the result will be close to the original s(a). In essence, such difficulties always arise when we are trying to find some data with the help of another measured quantity. At those times, we assume that if our measured quantity differs from the real quantity by a very small amount, then the value computed from it will also be very close to its actual value.