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Continuity & Differentiability Trick: Solve in 3 Steps

⚡ Maths Trick Why It Works 📅 2 September 2026
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Continuity & Differentiability Trick: Solve in 3 Steps, Maths Tricks · Why It Works | jijnas.com

Testing continuity and differentiability in piecewise functions does not require pages of long limit expansions. With a quick systematic check, you can determine if a function is continuous or differentiable at a critical point in under 30 seconds!

The Trick

  1. Check the Function Values for Continuity. Evaluate the left-hand piece and right-hand piece directly at the critical point x = c. If f(c-) = f(c+), the function is continuous at x = c.
  2. Differentiate Each Piece. Take the derivative of each defined interval piece independently to find the left-hand derivative function f'L(x) and right-hand derivative function f'R(x).
  3. Compare the Derivative Values. Evaluate both derivative pieces at x = c. If f'L(c) = f'R(c), the function is differentiable at x = c. If they differ, a sharp corner exists and it is non-differentiable.
Why it works: Continuity requires that the Left Hand Limit (LHL) equals the Right Hand Limit (RHL) equals the value at the point. Differentiability requires that the rate of change approaching from the left matches the rate of change approaching from the right. Differentiating the algebraic pieces directly computes these directional rates without re-evaluating the full difference quotient limit limh → 0 f(c+h)-f(c)h.

Examples

Example 1 (Easy): Is f(x) = begin{cases} 2x + 3, & x ≤ 1 5x, & x > 1 end{cases} continuous and differentiable at x = 1?

Continuity check at x = 1:

Left piece: 2(1) + 3 = 5
Right piece: 5(1) = 5
Since LHL = RHL = 5, f(x) is continuous at x = 1.

Differentiability check at x = 1:

Derivative of left piece: d/dx(2x + 3) = 2
Derivative of right piece: d/dx(5x) = 5
Since 2 ≠ 5, f(x) is continuous but NOT differentiable at x = 1.

Example 2 (Medium): Test differentiability of f(x) = begin{cases} x2, & x ≤ 2 4x - 4, & x > 2 end{cases} at x = 2.

Continuity check at x = 2:

Left piece: (2)^2 = 4
Right piece: 4(2) - 4 = 4
Since LHL = RHL = 4, f(x) is continuous at x = 2.

Differentiability check at x = 2:

Derivative of left piece: d/dx(x^2) = 2x → At x = 2: 2(2) = 4
Derivative of right piece: d/dx(4x - 4) = 4 → At x = 2: 4
Since Left Derivative = Right Derivative = 4, f(x) is differentiable at x = 2.

Example 3 (Hard): Find constants a and b so that f(x) = begin{cases} ax2 + 1, & x ≤ 1 3x + b, & x > 1 end{cases} is differentiable at x = 1.

Set continuous condition at x = 1:

a(1)^2 + 1 = 3(1) + b ⇒ a + 1 = 3 + b ⇒ a - b = 2

Set differentiable condition at x = 1:

d/dx(ax^2 + 1) = 2ax → At x = 1: 2a
d/dx(3x + b) = 3 → At x = 1: 3
2a = 3 ⇒ a = 3/2

Solve for b:

(3/2) - b = 2 ⇒ b = -1/2

Watch Out

Never test differentiability without checking continuity first. A discontinuous function can accidentally show equal left and right derivatives on paper, but it can never be differentiable if it is not continuous! Also watch out for modulus functions like f(x) = |x - 3|: they create sharp corners at their roots (x = 3), making them continuous everywhere but non-differentiable at those roots.

Where You'll Use This

1. Robotics and Motion Planning: Autonomous vehicles require continuous velocity and acceleration profiles to avoid instant jerky turns or physical stress on mechanical joints.

2. Computer Graphics and Animation: Smooth 3D curves (Bézier curves) use continuous derivatives to ensure light reflects smoothly across rendered surfaces without visual glitches.

3. Economics: Marginal cost and profit optimization rely on continuous, differentiable revenue curves to find true breakeven and maximum points.

Practice

1. Is f(x) = |x - 2| continuous at x = 2?

2. Is f(x) = |x - 2| differentiable at x = 2?

3. Check if f(x) = begin{cases} x3, & x ≤ 0 x2, & x > 0 end{cases} is differentiable at x = 0.

4. Find k if f(x) = begin{cases} kx + 1, & x ≤ 2 x2, & x > 2 end{cases} is continuous at x = 2.

5. Is f(x) = begin{cases} 3x - 1, & x < 1 x2 + 1, & x ≥ 1 end{cases} continuous at x = 1?

Answers: 1. Yes | 2. No | 3. Yes (Left deriv = 0, Right deriv = 0) | 4. k = 3/2 | 5. Yes (LHL = 2, RHL = 2)
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Discussion

0 be kind · most people here are students
No comments yet, be the first voice here. 🌱

🧭 Is it continuous? Is it differentiable? Decide in four checks

Answer each question, get to the fix. Take the same path you would at the bench.

See the whole map at once
  1. Is f(a) defined at the point x = a?Yes, f(a) existsNo, f(a) is not defined
  2. Do the left-hand limit and right-hand limit at x = a exist and equal each other?Yes, LHL = RHLNo, they differ or do not exist
  3. Does that common limit equal f(a)?Yes, limit = f(a)No, limit differs from f(a)
  4. f is continuous at a. Now the derivative: do LHD and RHD at x = a exist and equal each other?Yes, LHD = RHDNo, they differ (a corner or kink)
  5. Verdict → Not continuous at x = a, so it cannot be differentiable there either. In the exam, state which of the three continuity conditions failed.
  6. Verdict → Continuous but NOT differentiable at x = a, like |x| at 0. Write both one-sided derivatives to show they differ.
  7. Verdict → Continuous AND differentiable at x = a. Quote the common derivative value as f'(a).

🔍 The traps that eat marks in this chapter

MythIf a function is continuous at a point, it must be differentiable there.

RealityFalse. |x| is continuous at 0 but not differentiable there: the left derivative is -1, the right is +1. Differentiability implies continuity, never the other way around.

MythIf the limit of f(x) exists at x = a, the function is continuous at a.

RealityOnly if that limit also EQUALS f(a). A hole with the point moved elsewhere has a perfectly good limit and is still discontinuous.

MythA function defined by different formulas on two sides must be discontinuous at the joint.

RealityPiecewise does not mean broken. If both pieces meet at the same value and the one-sided limits agree, the joint is continuous, and often differentiable too.

MythIf f is differentiable, its derivative f' must be continuous.

RealityNot guaranteed. x^2 sin(1/x) (with value 0 at 0) is differentiable everywhere, yet f' jumps around near 0. Differentiable and continuously differentiable are different claims.

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