91Ó°ÊÓ

The formal definition of continuity is based on limits. A function \(f(x)\) is continuous at a point \(a\) if and only if the following three conditions are met: 1. \(f(a)\) is defined, 2. \(\lim_{x\to a}f(x)\) exists, 3. \(\lim_{x\to a}f(x) = f(a)\). This definition captures the idea of no holes or breaks in the graph, but it also provides a precise mathematical framework to understand and work with continuity.

Consider the function: $$ f(x) = \begin{cases} 1 & \text{if } x = 0 \\ x & \text{otherwise} \end{cases} $$ The graph of this function has a hole at \(x=0\): the value \(f(0)\) is \(1\), but the graph's overall trend appears to follow the line \(y=x\), with the point \((0,1)\) obviously disconnected from the rest of the curve. According to the informal definition of continuity (a function is continuous without holes or breaks), this function would be continuous at \(x=0\). However, using the formal definition of continuity, it is not. To use the formal definition, we check if the three conditions above are met at \(x=0\). First, we see that \(f(0)\) is defined and equals \(1\). Next, we calculate the limit of \(f(x)\) as \(x\) approaches \(0\). $$ \lim_{x\to 0}f(x) = \lim_{x\to 0} x = 0 $$ The limit exists and equals \(0\). However, the limit does not equal \(f(0)\), that is, \(\lim_{x\to 0}f(x) \neq 1\). Therefore, \(f(x)\) is *not* continuous at \(x=0\), even though the graph appears to have "no breaks" according to the informal definition. This example demonstrates that the informal definition of continuity cannot account for all cases where the proper concept of continuity is needed, and hence why it is not an adequate definition.