Continuity and Differentiability | Class 12 Mathematics Notes
By ConceptScroll Team · Published on 17 July 2026 · 2 min read
Continuity and Differentiability – this guide gives you a concise, exam-ready overview of Continuity and Differentiability from Class 12 Mathematics, written by ConceptScroll editors and reviewed against the latest NCERT textbook.
Differentiability
Differentiability is a stronger condition than continuity. A function f is said to be differentiable at a point x = a if the derivative f'(a) exists at that point. The derivative at a point measures the rate of change or slope of the function at that point.
Formally, f is differentiable at x = a if the limit of [f(a + h) - f(a)] / h as h approaches 0 exists. This limit, if it exists, is the derivative f'(a).
Differentiability implies continuity, but the converse is not always true. That is, if a function is differentiable at a point, it must be continuous there, but a continuous function need not be differentiable.
The NCERT textbook explains this with examples such as f(x) = |x|, which is continuous everywhere but not differentiable at x = 0 due to a sharp corner.
The section also discusses the geometric interpretation of differentiability as the existence of a tangent line at the point with a finite slope.
The concept of differentiability is foundational for calculus, enabling the study of rates of change, optimization, and motion.
📊 Diagram: Diagrams include graphs of functions with tangents at points where derivatives exist and graphs showing corners where derivatives do not exist.
🧪 Activity: Activity: Students compute derivatives at specific points to verify differentiability.
🔗 Connection: This section leads to exploring the relationship between continuity and differentiability in the next section.
Frequently asked questions
Which of the following is NOT a necessary condition for a function $f(x)$ to be continuous at $x = a$?
$f'(a)$ exists
If a function $f(x)$ has a jump discontinuity at $x = c$, which of the following statements is true?
$\lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x)$
State the three conditions for continuity of a function $f(x)$ at a point $x = a$.
The three conditions for continuity at $x = a$ are: 1) $f(a)$ is defined. 2) $\lim_{x \to a} f(x)$ exists. 3) $\lim_{x \to a} f(x) = f(a)$.
Explain with an example what is meant by removable discontinuity.
A removable discontinuity occurs at $x = a$ if the limit $\lim_{x \to a} f(x)$ exists but is not equal to $f(a)$ or $f(a)$ is not defined. For example, the function $f(x) = \frac{x^2 - 1}{x - 1}$ has a removable discontinuity at $x = 1$ because $f(1)$ is undefined but $\lim_{x \to 1} f(x) = 2$.
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