Discontinuities

Level FundamentalDifficulty ★★★★★Concept⌖ Open in the map

What is it?

Points where continuity fails: removable (the limit exists but disagrees with the value), jump (different one-sided limits) and essential (a one-sided limit does not exist).

Formulas

\lim_{x\to a^-} f(x) \ne \lim_{x\to a^+} f(x) \quad \text{(jump)}

Where it shows up in computing

Image processing and computer vision★★★★★frequentSignals, media and vision
Edges in an image are (approximate) jump discontinuities of intensity; edge detectors look for them.

Where it shows up in AI

Activation functions★★★★★historicalAI and machine learning
The perceptron's step activation is discontinuous; replacing it by the sigmoid made backpropagation possible.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

ℒ AI and machine learning

Activation functions★★★★★
Activation functions→Neural networks★★★★★
Activation functions→Neural networks→Backpropagation★★★★★
Activation functions→Neural networks→Loss landscape★★★★★
Activation functions→Neural networks→Backpropagation→Deep learning★★★★★
Activation functions→Neural networks→Loss landscape→Second-order (Hessian-based) optimization★★★★★
+3

∿ Signals, media and vision

Image processing and computer vision★★★★★
Image processing and computer vision→Media compression (JPEG, MP3, video)★★★★★

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.