NCERT CLASS 12 | Maths | Chapter 5 | exercise 5.1 | Question 26 to 34 | By Hit8OM

#Continuity​​​​ and Differentiability are fundamental concepts in calculus, typically covered in Class 12 mathematics.
1. Continuity:
A function is said to be continuous at a point if the following conditions are met:
The function is defined at that point.
The limit of the function as it approaches the point exists.
The value of the function at that point is equal to the limit.
In simpler terms, there is no "break" or "jump" in the graph of the function at the given point.
Mathematically, a function is continuous at if:
lim_{x \to c} f(x) = f(c)
Point continuity: Continuity at a specific point.
Interval continuity: Continuity over a range or interval.
2. Differentiability:
A function is differentiable at a point if the derivative exists at that point.
This implies the function must be continuous at that point, but not all continuous functions are differentiable.
A function is differentiable at if the following limit exists:
f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h}
If a function is differentiable at every point in its domain, it is said to be differentiable on that interval.
Key Concepts:
A function cannot be differentiable at a point where it is not continuous.
Functions like are continuous but not differentiable at some points (e.g., at ).
The concepts are used to analyze the smoothness of curves and the rate of change of quantities in physics, economics, and other fields.
In summary, continuity ensures the function has no abrupt changes at a point, while differentiability involves the function having a smooth and well-defined slope at that point
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