03 Continuity: Limits, Discontinuities, and Theorems Free Online FlashCards

What are the three conditions for continuity at x=a?

A function is continuous at x=a when f(a) is defined, lim_{x→a} f(x) exists, and lim_{x→a} f(x)=f(a).

How is continuity defined on a closed interval [a,b]?

For a closed interval [a,b], f must be continuous on (a,b), right-continuous at a, and left-continuous at b.

What characterizes a removable discontinuity?

A removable discontinuity occurs when a finite two-sided limit exists, but the function is undefined there or has a different value.

Where is every polynomial continuous?

Every polynomial is continuous at every real number because polynomials are defined everywhere and obey the continuity algebra rules.

Which algebraic operations preserve continuity?

If f and g are continuous at a, then f+g, f−g, cf, and fg are continuous at a; f/g is also continuous when g(a)≠0.

What is the relationship between differentiability and continuity?

Differentiability at a implies continuity at a, but continuity does not imply differentiability. For example, |x| is continuous but not differentiable at 0.

What does the Extreme Value Theorem guarantee?

If f is continuous on [a,b], it attains both an absolute maximum and an absolute minimum on that interval.

What does the Intermediate Value Theorem guarantee?

If f is continuous on [a,b] and N lies between f(a) and f(b), then some c∈[a,b] satisfies f(c)=N.

What is the root-existence form of the IVT?

If f is continuous on [a,b] and f(a)f(b)<0, then there is at least one c∈(a,b) with f(c)=0.

When is a composite function continuous at a?

If g is continuous at a and f is continuous at g(a), then (f∘g)(x)=f(g(x)) is continuous at a.

When is the inverse of a function continuous?

If f is continuous and one-to-one on an interval I, then f⁻¹ is continuous on the range f(I).

What condition defines a jump discontinuity?

A jump discontinuity occurs when both finite one-sided limits exist but are unequal: lim_{x→a−}f(x) ≠ lim_{x→a+}f(x).