Continuity and Differentiability
Class 12 · Mathematics
4 ideas in Class 12 Mathematics, explained in plain words, with games and quizzes to practise.
Continuity at a Point
Part of: Continuity and the Chain Rule
A function is continuous at a point when the left march, the right march and the actual value all agree. Jumps split the marches, holes move the value away, and integer-step functions break at every whole number. Polynomials march smoothly everywhere.
Why it matters. Continuity gates differentiability, and one-sided mismatches are the classic trap.
Formulas
- Continuous at c means limit at c equals value at c
- Polynomials are continuous on all reals
- Match left limit, right limit and value
Key words
- Continuity at a point
- : Left limit, right limit and value all matching there.
- Left-hand limit
- : The value approached from the smaller side.
- Right-hand limit
- : The value approached from the larger side.
- Removable discontinuity
- : A lone hole where only the value disagrees.
- Jump discontinuity
- : A break where the two one-sided limits differ.
Try it in two minutes
- March towards a point from both sides on a graph.
- Compare the two march ends with the dot at the point.
- Name the break as jump, hole or step.
Try at home. Test f(x) = (x² − 4)/(x − 2) at x = 2 for a hole.
Differentiability and Chain Rule
Part of: Continuity and the Chain Rule
Differentiability is stricter than continuity: a smooth slope must exist, so corners like |x| at 0 fail despite staying continuous. The chain rule multiplies outer and inner derivatives, implicit equations differentiate term by term, and inverse trig derivatives come from flipping sine, cosine and tangent.
Why it matters. Almost every later derivative chains, hides y inside equations, or inverts a trig function.
Formulas
- Differentiable forces continuous; converse fails
- Chain rule multiplies outer by inner derivative
- Derivative of tan⁻¹x is 1 over (1 + x²)
Key words
- Differentiability
- : Owning a well-defined slope at a point.
- Chain rule
- : Multiplying the outer derivative by the inner derivative.
- Implicit function
- : A relation mixing x and y without isolating y.
- Composite function
- : One function nested inside another.
Try it in two minutes
- Check the corner of |x| for a missing slope.
- Peel a composite function into outer and inner layers.
- Flip sin⁻¹ into sine and solve for the slope.
Try at home. Differentiate cos(x²) and compare with differentiating cos²x.
Exponential, Log and Clever Differentiation
Part of: Exponentials, Parameters and Higher Order
The natural exponential reproduces itself on differentiation while log x differentiates to 1/x, with log meaning base e throughout. Chaining wraps these rules around any inner function, and log-first differentiation unlocks variable powers like xˣ plus long products and quotients.
Why it matters. Growth, decay and messy products all differentiate through these two rules.
Formulas
- Derivative of eˣ is eˣ; derivative of log x is 1 over x
- Log first for powers like xˣ
- log x here means base e
Key words
- Natural exponential function
- : e to the x, cloning itself on differentiation.
- Natural logarithm
- : Log to base e, written log x in this chapter.
- Logarithmic differentiation
- : Taking logs first, then differentiating.
- Base e
- : The natural growth constant between 2 and 3.
Try it in two minutes
- Differentiate e⁻³ˣ by chaining the exponent.
- Differentiate cos(log x) layer by layer.
- Take logs of xˣ, then differentiate.
Try at home. Differentiate e to the sin x and log(sin x) side by side.
Parametric and Second Order
Part of: Exponentials, Parameters and Higher Order
When x and y each ride on a parameter t, the slope dy/dx divides dy/dt by dx/dt, provided dx/dt is nonzero. Differentiating a first derivative again gives the second order derivative, which tracks how the slope itself bends.
Why it matters. Motion paths arrive parametrically, and second derivatives reveal acceleration and curvature.
Formulas
- Parametric dy/dx is (dy/dt) over (dx/dt)
- Second derivative repeats differentiation on f prime
- Keep dx/dt nonzero
Key words
- Parameter
- : A third variable carrying both x and y.
- Parametric form
- : Writing x and y each as functions of t.
- Second-order derivative
- : The derivative of the first derivative.
- Order of a derivative
- : How many times differentiation repeats.
Try it in two minutes
- Differentiate x and y separately against t.
- Divide the y-rate by the x-rate.
- Differentiate the result once more for second order.
Try at home. Find dy/dx for x = 4t and y = 5t² without removing t.
What your child will practise
- Continuity at a Point: Jungle Puzzle (Clue Puzzle)
- Continuity at a Point: Space Quick Fire (Quick Fire)
- Differentiability and Chain Rule: Candy Puzzle (Clue Puzzle)
- Exponential, Log and Clever Differentiation: Candy Quest (Flip and Match)
- Parametric and Second Order: Candy Quick Fire (Quick Fire)
- Quiz: Continuity at a Point (8 questions)
- Quiz: Differentiability and Chain Rule (8 questions)
- Quiz: Exponential, Log and Clever Differentiation (8 questions)
- Quiz: Parametric and Second Order (8 questions)