DIFFERENTIABILITY - Mathematical Definition

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This educational video provides a focused lecture on the concept of differentiability in calculus, presented by instructor Yash Jain Sir from Knowledge Gate. The session begins with a formal mathematical definition of differentiability at a specific point, establishing the necessary limit condition. A central theme of the lecture is the logical relationship between differentiability and continuity. The instructor explicitly states that while differentiability guarantees continuity, the reverse is not true. To substantiate this claim, he performs a detailed analysis of the absolute value function, f(x) = |x|, demonstrating through limit calculations that a function can be continuous at a point yet fail to be differentiable due to a sharp corner or cusp. The lecture effectively bridges the gap between abstract definitions and practical application.

Chapters

1. 0:00 – 2:00 00:00-02:00

   The video opens with a title card featuring the word CALCULUS surrounded by various mathematical formulas. The instructor, identified as Yash Jain Sir from Knowledge Gate, appears in front of a whiteboard titled Differentiability -> Mathematical Definition. He writes the formal definition: a function f: R -> R is differentiable at x=a if f'(a) exists. He expands this to show the limit definition: lim h->0 [f(a+h) - f(a)] / h must exist. Below this, he lists two critical implications: (1) if a function is differentiable, it is continuous, and (2) if a function is continuous, it is not necessarily differentiable, indicated by a crossed arrow. He underlines x=a and f'(a) to emphasize the point of interest.

2. 2:00 – 5:00 02:00-05:00

   The instructor moves to a concrete example to illustrate the second point. He writes f(x) = |x| and defines it piecewise as x for x >= 0 and -x for x < 0. He asserts that the function is continuous at x=0 but not differentiable. He sets up the derivative limit at x=0: lim h->0 [f(0+h) - f(0)] / h. He splits this into Left Hand Limit (LHL) and Right Hand Limit (RHL). He calculates the RHL as lim h->0+ h/h = 1 and the LHL as lim h->0- -h/h = -1. He draws a V-shaped graph of |x| on the right side of the board to visualize the sharp corner at the origin.

3. 5:00 – 5:44 05:00-05:44

   The instructor concludes the proof by highlighting that since the LHL (-1) is not equal to the RHL (1), the limit does not exist, and thus the function is not differentiable at x=0. He writes LHL != RHL in red ink to emphasize the inequality. He also writes LD = RD with a checkmark to show the general condition for differentiability, contrasting it with the current case where LD != RD. He circles the limit expression and the result -1 to draw attention to the specific values. The video concludes with a black screen displaying THANKS FOR WATCHING.

The lecture successfully demystifies the relationship between continuity and differentiability. By starting with the rigorous limit definition and immediately applying it to a classic counterexample, the instructor provides a clear, visual, and mathematical proof that continuity is a prerequisite but not a guarantee for differentiability. The step-by-step calculation of one-sided limits for the absolute value function serves as a practical template for students to analyze other non-differentiable points, reinforcing the concept through calculation of one-sided limits.