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• Real Analysis
• 1. Sets and Relations
• 2. Infinity and Induction
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 6.1. Limits
• 6.2. Continuous Functions
• 6.3. Discontinuous Functions
• 6.4. Topology and Continuity
• 6.5. Differentiable Functions
• 6.6. A Function Primer
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
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Theorem 6.3.6: Discontinuities of Monotone Functions

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If f is a monotone function on an open interval (a, b), then any discontinuity that f may have in this interval is of the first kind.

If f is a monotone function on an interval [a, b], then f has at most countably many discontinuities.

Context

Proof:

Suppose, without loss of generality, that f is monotone increasing, and has a discontinuity at x₀. Take any sequence x_n that converges to x₀ from the left, i.e. x_n x₀. Then f( x_n) is a monotone increasing sequence of numbers that is bounded above by f(x₀). Therefore, it must have a limit. Since this is true for every sequence, the limit of f(x) as x approaches x₀ from the left exists. The same prove works for limits from the right.

Note: This proof is actually not quite correct. Can you see the mistake ? Is it really true that if x_n converges to x₀ from the left then f(x_n) is necessarily increasing ? Can you fix the proof so that it is correct ?

As for the second statement, we again assume without loss of generality that f is monotone increasing. Define, at any point c, the jump of f at x = c as:

j(c) = f(x) - f(x)

Note that j(c) is well-defined, since both one-sided limits exist by the first part of the theorem. Since f is increasing, the jumps j(c) are all non-negative. Note that the sum of all jumps can not exceed the number f(b) - f(a). Now let J(n) be the set of all jumps c where j(c) is greater than 1/n and let J be the set of all jumps of the function in the interval [a, b]. Since the sum of jumps must be smaller than f(b) - f(a), the set J(n) is finite for all n. But then, since the union of all sets J(n) gives the set J, the number of jumps is a countable union of finite sets, and is thus countable.