BeginnerTopic 26 of 104Engineering Mathematics
Continuity
A function is continuous if small input changes cause small output changes with no jumps.
In depth
A function is continuous at a if the limit exists, f(a) exists and they are equal. Continuous functions on closed intervals attain maxima and minima and satisfy the intermediate value theorem. Discontinuities may be jumps, holes or infinite.
Key points
- Polynomials and sin/cos are continuous everywhere
- Step inputs are discontinuous signals
- Differentiable implies continuous
ExampleThe intermediate value theorem guarantees a root of x³-x-1 between 1 and 2.
What is Continuity?
A function is continuous if small input changes cause small output changes with no jumps.
Why is it important to understand continuity?
Continuity guarantees that small input changes give small output changes, which is needed for many theorems and for stable numerical methods.
What are the three conditions for continuity at a point?
The function must be defined at the point, the limit must exist there, and the limit must equal the function value.
Is a step input continuous?
No. A step jumps instantly, so it is discontinuous, though its integral, a ramp, is continuous.
How is Continuity used in practice?
Typical applications include checking a piecewise material model at its transition point, confirming that a control signal has no jumps before differentiating it and testing whether a sensor curve can be interpolated safely.
What should I learn before Continuity?
Study "Limits" first (topic 25 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Continuity?
Polynomials and sin/cos are continuous everywhere. Step inputs are discontinuous signals. Differentiable implies continuous.
Can you give an example of Continuity?
The intermediate value theorem guarantees a root of x³-x-1 between 1 and 2.
What comes after Continuity?
The next topic is "Derivatives", which builds directly on this one.
How difficult is Continuity?
It is classed as Beginner in Engineering Mathematics (topic 26 of 104). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.