Continuity and Differentiability (Class 12 Maths): The Checks That Actually Score
Most marks in this chapter are lost not on the calculus but on failing to test the one point where the definition of the function changes.
The definition, written the way the answer should be written
A function f is continuous at x = a when all three of the following exist and are equal:
Notice that f(a) must be defined. A function with a hole at a point is not continuous there even if both one-sided limits agree perfectly, and questions are set specifically to catch that.
Differentiability at x = a requires the two one-sided derivatives to exist and be equal:
The standard question: a piecewise function with an unknown
A typical board question gives a function defined in two pieces with a constant k, and asks for the value of k making it continuous. The method never changes.
- Identify the single point where the definition changes. Away from that point each piece is a polynomial, rational, trigonometric or exponential function and is continuous automatically — say this in a sentence rather than proving it.
- Compute the left-hand limit using the piece valid to the left.
- Compute the right-hand limit using the piece valid to the right.
- Write down f(a) from whichever piece defines it.
- Set all three equal and solve for the unknown.
The mark scheme awards the two one-sided limits separately. A solution that computes only one limit and equates it to f(a) loses those marks even when the final answer is correct.
The three function types that appear again and again
| Function | Continuous? | Differentiable? | Where it fails |
|---|---|---|---|
| |x| | Everywhere | Not at x = 0 | Lf′ = –1, Rf′ = +1 |
| |x – a| | Everywhere | Not at x = a | Corner at x = a |
| [x], greatest integer | Not at integers | Not at integers | Jump of 1 at each integer |
| x – [x], fractional part | Not at integers | Not at integers | Jump at each integer |
| x sin(1/x), f(0) = 0 | Everywhere | Not at x = 0 | Derivative limit oscillates |
| x² sin(1/x), f(0) = 0 | Everywhere | Everywhere | Squeeze theorem gives f′(0) = 0 |
Differentiation techniques the chapter then requires
Chain rule and composite functions
Differentiate the outside function, keep the inside unchanged, then multiply by the derivative of the inside. Write the intermediate substitution explicitly for a nested function; skipping it is where sign errors enter.
Logarithmic differentiation
Use it whenever the variable appears in the exponent, or when the expression is a long product or quotient. For y = xx, no ordinary rule applies, since neither the power rule nor the exponential rule fits.
Parametric and implicit forms
For parametric equations, dy/dx = (dy/dt) ÷ (dx/dt). For the second derivative the common error is to differentiate dy/dx with respect to t and stop; you must divide by dx/dt again.
Rolle’s and Lagrange’s theorems
Both are stated with hypotheses that must be verified before use, and the verification carries marks. Rolle requires continuity on the closed interval, differentiability on the open interval, and equal values at the endpoints; the conclusion is that f′(c) = 0 for some c inside. Lagrange drops the equal-endpoint condition and concludes that f′(c) equals the average rate of change.
Questions often supply a function that fails one hypothesis, such as |x| on an interval containing zero, and ask whether the theorem applies. The expected answer names the specific condition that fails.
Frequently asked questions
If a function is continuous, is it differentiable?
No. Continuity is necessary but not sufficient. The function |x| is continuous at x = 0 but has left derivative –1 and right derivative +1, so it is not differentiable there. The implication runs only from differentiability to continuity.
How many marks does showing both one-sided limits actually carry?
In the standard CBSE scheme the left-hand limit and the right-hand limit are each credited separately, with a further mark for equating them to f(a) and solving. A solution that jumps straight to the answer typically loses those working marks even when the value is right.
Why is x sin(1/x) continuous but not differentiable at zero?
The function value is squeezed to zero as x approaches zero, so it is continuous if f(0) is defined as 0. The difference quotient, however, reduces to sin(1/h), which oscillates between –1 and 1 and has no limit, so the derivative does not exist.
When should I use logarithmic differentiation?
Whenever the variable appears in an exponent, as in x^x or (sin x)^x, and whenever the expression is a long product or quotient where taking logarithms turns multiplication into addition and makes the differentiation manageable.
What is the difference between Rolle’s theorem and the mean value theorem?
Rolle is the special case in which f(a) = f(b), so the guaranteed derivative is zero. Lagrange removes that restriction and guarantees a point where the derivative equals the average rate of change across the interval.
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