Mathematics
High School (11–12)
Functions and Calculus › Continuity and DifferentiabilityChapter 9Contains Standard contentContinuity and Differentiability
Draw the graph of a bus journey without lifting your pencil. It is smooth where the bus speeds up gently, and it has a sharp corner where the driver brakes hard. This chapter turns that picture into mathematics. You test when a function is continuous, find where a graph has a gap or a corner, learn every rule of differentiation, and meet Rolle's theorem and the mean value theorem. It covers the unit of the Indian syllabus (NCERT, JEE Main) and the matching topics of A-Level, IB and AP. Python checks every limit and every derivative on the computer.
Part of Mathematics · High School (11–12) (chapter 9)
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Continuity of Functions
A graph with no break is continuous, and a graph with a gap is not. Turn that picture into a test with left-hand and right-hand limits, name the three kinds of break, and check piecewise functions. Then use the Intermediate Value Theorem to show that an equation has a root. It covers the first part of the Indian syllabus unit (NCERT, JEE Main) and the matching topic of A-Level, IB and AP.
Differentiability and the Rules of Differentiation
Where does a graph have a corner? Define the derivative with limits, see why a differentiable function must be continuous, and then learn the rules: the chain rule, inverse trigonometric, exponential and logarithmic functions, and implicit, logarithmic and parametric differentiation. It covers the Indian syllabus (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.