Continuity and Differentiability Made Simple
Ever wondered why a roller‑coaster can glide smoothly in one spot but suddenly feel a sharp jolt?
💡 In Simple Words: A function is continuous if you can draw its graph without lifting your pencil. If you can also slide a straight line (the tangent) that just kisses the curve at a point, the function is differentiable there.
What does “continuous” really mean?
A function is continuous at a point c when three things happen:
- The function has a value at c (you can plug c into the formula).
- The limit (what the function is trying to become as you get really close to c) exists.
- The limit equals the actual value f(c).
Think of a water pipe. If the pipe has no holes and the water flows right up to the faucet, that’s continuity. The water doesn’t suddenly disappear.
Example: f(x)=x² is continuous everywhere because you can approach any x‑value, the limit exists, and it matches the function’s value.
When is a function differentiable?
A function is differentiable at c if you can draw a single straight line that just touches the curve at c without crossing it. That line is called the tangent, and its slope is the derivative (instant rate of change).
Imagine a bike rider gliding on a smooth road. The rider’s speed at an exact instant is the derivative. If the road has a sharp corner, you can’t define a single instant speed – that’s a non‑differentiable point.
Example: f(x)=x² is differentiable everywhere (its derivative is 2x). In contrast, f(x)=|x| (absolute value) has a “V” shape at 0, so it isn’t differentiable there.
Key relationship between continuity and differentiability
Every differentiable function is automatically continuous, but a continuous function need not be differentiable.
Analogy: A smooth road (differentiable) is certainly free of holes (continuous). A road with a small bump is still drivable (continuous) but you can’t roll a perfect tangent wheel over the bump.
How to test continuity
Follow these steps at the point c you’re interested in:
- Find the left‑hand limit limₓ→c⁻ f(x).
- Find the right‑hand limit limₓ→c⁺ f(x).
- Check the function value f(c).
- If the two limits exist, are equal, and equal f(c), the function is continuous at c.
How to test differentiability
Once continuity is confirmed, test the slopes:
- Compute the left‑hand derivative limₕ→0⁻ [f(c+h)–f(c)]/h.
- Compute the right‑hand derivative limₕ→0⁺ [f(c+h)–f(c)]/h.
- If both one‑sided derivatives exist and are equal, the function is differentiable at c.
Worked example: f(x)=|x|
Step 1 – continuity
Left limit as x→0⁻ |x| = 0, right limit as x→0⁺ |x| = 0, and f(0)=0. All match, so the function is continuous at 0.
Step 2 – differentiability
Left‑hand derivative: limₕ→0⁻ [|0+h|–0]/h = limₕ→0⁻ (–h)/h = –1.
Right‑hand derivative: limₕ→0⁺ [|0+h|–0]/h = limₕ→0⁺ h/h = 1.
The one‑sided slopes are different, so |x| is not differentiable at 0.
Quick comparison
| Aspect | Continuity | Differentiability |
|---|---|---|
| What it checks | No “breaks” in the graph | Existence of a single tangent line |
| Mathematical condition | limₓ→c⁻ f(x)=limₓ→c⁺ f(x)=f(c) | limₕ→0⁻ [f(c+h)–f(c)]/h = limₕ→0⁺ [f(c+h)–f(c)]/h |
| Typical example | f(x)=x³ (continuous everywhere) | f(x)=x³ (differentiable everywhere) |
| Counter‑example | f(x)=|x| (continuous but not differentiable at 0) | f(x)=|x| at x=0 (fails differentiability) |
| Implication | None | Differentiable ⇒ continuous |
📝 Likely Exam Questions
- State the definition of continuity at a point and give one example of a continuous function.
Answer: A function f is continuous at c if limₓ→c f(x)=f(c). Example: f(x)=x². - Explain why a differentiable function must be continuous. Provide a short reasoning.
Answer: Differentiability gives a finite limit for the difference quotient from both sides, which forces the left‑ and right‑hand limits to equal f(c). Hence the function cannot have a jump or hole. - Test the function f(x)=√x for continuity and differentiability at x=0.
Answer: f(0)=0, limₓ→0⁺ √x=0, left limit does not exist (domain restriction), so continuity holds on the right only. Derivative f'(x)=1/(2√x) blows up at 0, so not differentiable there. - Find the points where f(x)=x|x| is differentiable.
Answer: f(x)=x² for x≥0 and –x² for x - Give a brief example where a function is continuous but not differentiable, and explain why.
Answer: f(x)=|x| is continuous at 0 because both side limits equal 0, but the left‑hand slope is –1 and right‑hand slope is 1, so no single tangent exists.