From the Left and Right
Sometimes a function does different things depending on which direction you approach from. We use notations \(x \to a^+\) (from the right) and \(x \to a^-\) (from the left).
Continuity: A function is "Continuous" at a point if the left limit, the right limit, and the actual value are all the same.
Worked Examples
Example 1: The Step Function
A function is 0 for \(x < 0\) and 1 for \(x \geq 0\).
- \(\lim_{x \to 0^-} f(x) = 0\).
- \(\lim_{x \to 0^+} f(x) = 1\).
- Because they are different, the "General Limit" at 0 does not exist. The function is discontinuous.
Example 2: Absolute Value
Find the limit of \(|x|/x\) as \(x \to 0\).
- From the right (\(x>0\)): \(x/x = 1\).
- From the left (\(x<0\)): \(-x/x = -1\).
- Limit does not exist.
The Bridge to Quantum Mechanics
In Quantum Mechanics, the wavefunction \(\psi(x)\) must be continuous. If there was a "jump" in the wavefunction, it would imply that the particle had an infinite momentum at that spot, which is physically impossible. When we solve for a particle meeting a wall (Chapter 11), we use these limit rules to "match" the wave on the left to the wave on the right. This "Matching Condition" is what determines which energy levels are allowed. Continuity is the physical requirement that prevents nature from being broken.