= Single-point changes preserve Riemann integrals
Changing the value of a bounded <Riemann-integrable function> at one point preserves its <Riemann integral>. The difference is supported at that point: partitions isolating it in intervals of arbitrarily small total length make both <Darboux sums> tend to zero. Thus a nonnegative single-point spike can have zero integral without vanishing everywhere. Its indefinite integral is identically zero, so the <fundamental theorem of calculus> derivative identity can fail at the changed point despite differentiability of the primitive.
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