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Properties of Continuous Functions

Properties of Continuous Functions

Continuous functions exhibit smooth, uninterrupted behavior, and combining them through standard operations retains this continuity. If f and g are continuous at a point a, then the functions f+g, f-g, cf (where c is a constant), fg, and fg (provided g(a)a) are also continuous at a. This allows the construction of complex functions from simpler continuous parts without losing smoothness.Polynomials, which are expressions formed by sums of powers of x with constant coefficients, are continuous...

Continuity of a Function

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A function is continuous at a point a if three conditions are met: the function is defined at a, the limit of the function as x approaches a exists, and this limit equals the function’s value. Mathematically, this is written asThis definition ensures the graph of the function does not exhibit any breaks, holes, or jumps at that point. Discontinuities occur when any of these conditions fail. A removable discontinuity exists when the two-sided limit exists but the function is either undefined or...

Continuity Equation

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The continuity equation asserts that the mass flow rate must remain constant for a steady flow of an incompressible fluid within a confined system. This principle applies to systems where fluid passes through varying cross-sectional areas, such as nozzles, syringes, and pipes.
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Continuity Equation

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Entropy Change in Reversible Processes

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Related Experiment Video

Updated: Jun 5, 2026

Reservoir Condition Pore-scale Imaging of Multiple Fluid Phases Using X-ray Microtomography

Reservoir Condition Pore-scale Imaging of Multiple Fluid Phases Using X-ray Microtomography

Published on: February 25, 2015

Explosive percolation transition is actually continuous.

R A da Costa¹, S N Dorogovtsev, A V Goltsev

¹Departamento de Física da Universidade de Aveiro, I3N, 3810-193 Aveiro, Portugal.

Physical Review Letters

|January 15, 2011

Summary

This summary is machine-generated.

Explosive percolation, initially thought to be discontinuous, is actually a continuous phase transition. This study reveals its unique scaling properties and critical exponents, challenging previous understandings of percolation theory.

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Research and Development of High-performance Explosives

Research and Development of High-performance Explosives

Published on: February 20, 2016

Area of Science:

Statistical Physics
Complex Systems
Network Science

Background:

A novel "explosive percolation" transition was recently reported for irreversible systems, contrasting sharply with ordinary percolation phenomena.
This discontinuous transition raised questions about the fundamental nature of phase transitions in complex networks.

Purpose of the Study:

To investigate the nature of the explosive percolation transition in a representative model.
To determine if the transition is indeed discontinuous or exhibits continuous behavior.
To characterize the scaling properties and critical exponents of this transition.

Main Methods:

Analysis of a representative model exhibiting explosive percolation.
Investigation of the system's behavior near the critical point.
Calculation of critical exponents and dimensions associated with the transition.

Main Results:

The explosive percolation transition was found to be a continuous, second-order phase transition.
The transition exhibits uniquely small critical exponents, particularly for the percolation cluster size.
Unusual scaling properties characteristic of this continuous transition were identified and described.

Conclusions:

The explosive percolation transition is reclassified as a continuous phase transition, distinct from initial reports.
The study elucidates the unique scaling behavior and critical exponents, providing a deeper understanding of percolation phenomena.
Findings challenge existing theories and offer new insights into phase transitions in irreversible complex systems.