How to Prove Continuity by Examining Domain Subsets in Calculus

What We're Solving:

We need to determine whether proving that a function f is continuous at a point x₀ when we restrict its domain to a subset E is the same as proving continuity on the original, larger domain D. This is a fundamental question about how domain restrictions affect continuity!

The Approach:

Think of this like asking: "If I can drive smoothly on my neighborhood streets, does that guarantee I can drive smoothly on ALL roads?" We need to understand what continuity really means and how the domain affects our ability to approach a point. The key insight is that continuity depends on how we can approach x₀, and a smaller domain might give us fewer ways to approach that point.

Step-by-Step Solution:

Step 1: Recall the definition of continuity A function f: D → ℝ is continuous at x₀ if for every ε > 0, there exists δ > 0 such that:

• For all x ∈ D with |x - x₀| < δ, we have |f(x) - f(x₀)| < ε

Notice how the definition explicitly mentions "for all x ∈ D" - the domain matters!

Step 2: Consider what happens with the restricted domain If we restrict f to E ⊂ D, we get a new function g: E → ℝ where g(x) = f(x) for all x ∈ E. Continuity of g at x₀ means: for every ε > 0, there exists δ > 0 such that:

• For all x ∈ E with |x - x₀| < δ, we have |g(x) - g(x₀)| < ε

Step 3: Compare the two conditions Here's the crucial observation: The continuity condition for the restricted function only needs to work for points in E, while continuity on D needs to work for ALL points in D that are close to x₀.

Step 4: Construct a counterexample Let's create a concrete example:

• Let D = [-1, 1] and E = [-1, 0] ∪ [0, 1] (so E = D in this case, but let's modify...)
• Actually, let E = {0} ∪ [1/2, 1]
• Define f(x) = 0 for x ∈ E
• But on the full domain D, define f(x) = 1 for x ∈ (0, 1/2)

Now f restricted to E is continuous at x₀ = 0 (since the only points in E near 0 is just {0} itself), but f is not continuous at 0 on the full domain D because points in (0, 1/2) are close to 0 but f jumps to value 1 there.

Step 5: Understand the general principle The restriction to E might "hide" points that would reveal discontinuity. If E doesn't include all points near x₀ that are in D, we might miss problematic behavior.

The Answer:

No, proving continuity on a subset E is NOT sufficient to prove continuity on the entire domain D.

The restriction to a subset can hide discontinuous behavior by excluding points that would demonstrate the discontinuity. Continuity depends on the behavior of the function at ALL points in the domain that are sufficiently close to x₀, not just some of them.

However, the converse is true: if f is continuous at x₀ on domain D, then any restriction of f to a subset E ⊂ D containing x₀ will also be continuous at x₀.

Memory Tip:

Think of it as "You can't prove you're a good driver by only driving on easy roads!" A smaller domain might exclude the "difficult points" that would reveal discontinuous behavior. Continuity requires good behavior everywhere in the domain near your point, not just in carefully chosen subsets.