Proving Continuity with Derivatives- Mathematical Guide
What "Differentiability Implies Continuity" Actually Means
Here's the deal: if a function has a derivative at some point, that function is automatically continuous at that point. No exceptions, no edge cases to worry about in standard calculus.
This is one of the cleanest theorems you'll encounter. It gives you a shortcut for proving continuity — skip the epsilon-delta work and just verify the derivative exists.
The Formal Statements You Need
Definition of Continuity at a Point
A function f(x) is continuous at x = a when three conditions hold:
- The function value f(a) exists
- The limit lim(x→a) f(x) exists
- Those two values are equal
That's it. That's the whole definition.
Definition of Differentiability at a Point
A function f(x) is differentiable at x = a when this limit exists:
f'(a) = lim(h→0) [f(a+h) - f(a)] / h
The limit must be the same whether you approach from the left or right. If that limit exists, you get a derivative.
The Theorem and Its Proof
Theorem: If f'(a) exists, then f is continuous at x = a.
Proof:
Start with what you know exists: f'(a). Write the limit definition:
f'(a) = lim(h→0) [f(a+h) - f(a)] / h
Multiply both sides by h (which approaches 0):
f'(a) · h = lim(h→0) [f(a+h) - f(a)]
As h → 0, the left side becomes f'(a) · 0 = 0
So: lim(h→0) [f(a+h) - f(a)] = 0
Rearrange: lim(h→0) f(a+h) - f(a) = 0
Which means: lim(h→0) f(a+h) = f(a)
That's the definition of continuity at x = a. ✓
The proof is straightforward. The derivative's existence forces the function value and the limit to match — you can't have one without the other.
What This Doesn't Go the Other Way
Here's where students get burned: continuity does NOT guarantee differentiability.
Classic example: f(x) = |x| at x = 0
- It's continuous at x = 0 (you can draw it without lifting your pen)
- It's NOT differentiable at x = 0 (the slope jumps — no single tangent line exists)
Other functions that are continuous but not differentiable everywhere:
- Absolute value functions at their corners
- Functions with vertical tangents
- The Weierstrass function (continuous everywhere, differentiable nowhere)
The implication flows in one direction only: derivative existence → continuity.
How to Actually Use This
Strategy: Prove Continuity Without Epsilon-Delta
When you need to show a function is continuous at a point:
- Find the derivative at that point
- If the derivative exists, you're done — continuity follows automatically
Example: Prove f(x) = x³ is continuous at x = 2.
Find the derivative: f'(x) = 3x²
f'(2) = 3(2)² = 12
Since f'(2) exists, f(x) is continuous at x = 2. Done.
No limit evaluation, no epsilon arguments. Just differentiation.
Quick Reference Table
| Property | Implication | Example |
|---|---|---|
| f is differentiable at a | f is continuous at a | f(x) = x² at x = 1 |
| f is continuous at a | f may or may not be differentiable at a | f(x) = |x| at x = 0 |
| f is differentiable everywhere | f is continuous everywhere | f(x) = sin(x) |
| f is continuous everywhere | NOT differentiable everywhere | f(x) = |sin(x)| |
Common Mistakes
- Assuming the reverse: Students see "derivative → continuous" and assume continuous → derivative. Wrong. The absolute value function destroys this assumption.
- Forgetting domain issues: The theorem only applies where the derivative exists. A function can fail to be differentiable at points outside your immediate concern.
- Mixing up definitions: Memorizing the theorem without understanding the proof means you won't catch subtle cases where it doesn't apply.
When to Use the Epsilon-Delta Approach Instead
The derivative shortcut works when you're dealing with nice, standard functions — polynomials, trig functions, exponentials, anything with straightforward derivatives.
Use the formal definition of continuity when:
- You're working with piecewise functions at boundary points
- The limit is tricky or involves indeterminate forms
- Your instructor requires the full proof
- You're dealing with functions that don't have elementary antiderivatives
For most calculus problems, checking the derivative's existence is faster and cleaner.
The Bottom Line
If you can find the derivative at a point, you automatically have continuity there. This theorem exists precisely so you don't have to redo work that's already been done.
Use it. It's a tool. Differentiate → state the derivative exists → cite the theorem → done.