Continuity Theorems: quick revision, 15 bites

What Continuity Really Means

No jumps, no breaks, no holes.

A function $f(x)$ is **continuous** at $x = a$ if three conditions hold: $f(a)$ is defined, $\lim_{x \to a} f(x)$ exists, and $\lim_{x \to a} f(x) = f(a)$. Intuitively, you can draw the graph without lifting your pen.

Takeaway. Continuity at a point means the limit and the function value agree.

Polynomials and Rationals

Polynomials are always continuous.

**Polynomial functions** are continuous everywhere on $\mathbb{R}$. **Rational functions** (ratios of polynomials) are continuous wherever the denominator is nonzero. This is a fundamental fact used in many problems.

Takeaway. Use polynomial continuity to quickly check continuity on intervals.

Algebra of Continuous Functions

Combine continuous functions, stay continuous.

If $f$ and $g$ are continuous at $a$, then so are $f+g$, $f-g$, $fg$, and $f/g$ (provided $g(a) \neq 0$). This allows building complex functions from simple ones.

Takeaway. Use algebraic operations to verify continuity of combined functions.

Composition of Continuous Functions

Compose continuous, stay continuous.

If $f$ is continuous at $a$ and $g$ is continuous at $f(a)$, then the **composition** $g \circ f$ is continuous at $a$. This is powerful for nested functions like $\sin(x^2)$.

Takeaway. Composition preserves continuity, enabling analysis of complex functions.

Intermediate Value Theorem (IVT)

Continuous functions hit every value between.

If $f$ is continuous on $[a,b]$ and $k$ is any value between $f(a)$ and $f(b)$, then there exists at least one $c \in (a,b)$ such that $f(c) = k$. This theorem is used to prove existence of roots.

Takeaway. IVT guarantees existence of solutions to equations on intervals.

Solving Equations with IVT

Prove a root exists without solving.

To show $f(x)=0$ has a root on $[a,b]$, check that $f$ is continuous and that $f(a)$ and $f(b)$ have opposite signs. Then IVT guarantees at least one $c$ with $f(c)=0$.

Takeaway. IVT is a non-constructive way to prove existence of solutions.

Forgetting to Check Continuity

IVT requires continuity first.

A common error is applying IVT to a function that is not continuous on the interval. Always verify continuity before using IVT.

Takeaway. Always check continuity before invoking IVT.

Extreme Value Theorem (EVT)

Continuous on closed interval, max and min exist.

If $f$ is continuous on a **closed interval** $[a,b]$, then $f$ attains both a **maximum** and a **minimum** value on that interval. This is essential for optimization problems.

Takeaway. EVT ensures that continuous functions on closed intervals have extrema.

Finding Extrema in Practice

Use EVT to locate max and min.

To find extrema of a continuous function on $[a,b]$, evaluate $f$ at critical points (where derivative is zero or undefined) and at endpoints. The largest value is the maximum, the smallest is the minimum.

Takeaway. EVT justifies the standard procedure for finding extrema on closed intervals.

Continuity on Intervals

Continuity on open vs closed intervals.

A function is **continuous on an open interval** if it is continuous at every point in that interval. On a closed interval $[a,b]$, we require continuity on $(a,b)$ and one-sided continuity at endpoints.

Takeaway. Interval continuity extends point continuity with endpoint conditions.

Ignoring Domain Restrictions

Rational functions have holes.

When checking continuity of rational functions, remember that they are undefined where the denominator is zero. Those points are not in the domain, so continuity is only considered where defined.

Takeaway. Always consider the domain before discussing continuity.

Step-by-Step Continuity Check

Follow a systematic procedure.

Takeaway. Use this checklist to solve continuity problems reliably.

Remember IVT and EVT

IVT for roots, EVT for extrema.

**I**ntermediate **V**alue **T**heorem: continuous functions take all intermediate values. **E**xtreme **V**alue **T**heorem: continuous functions on closed intervals have max and min.

Takeaway. Associate IVT with roots and EVT with optimization.

Continuity and Differentiability

Differentiable implies continuous.

If a function is **differentiable** at a point, it is automatically continuous there. The converse is not true: $f(x)=|x|$ is continuous but not differentiable at $x=0$.

Takeaway. Differentiability is stronger than continuity.

Problem-Solving Workflow

A general strategy for continuity problems.

Takeaway. Follow this workflow to tackle any continuity problem.