Question

The function $f(x) = \begin{cases} x^2 & \quad \text{for } x < 1\\ 2 - x & \quad \text{for } x \geq 1 \end{cases}$ is

• A. not differentiable at $x = 1$
• B. differentiable at $x = 1$
• C. not continuous at $x = 1$
• D. none of these

Answer 1

The Correct Option is A

$f(x) =
\begin{cases}
x^2 & \quad \text{for } x < 1\\
2 - x & \quad \text{for } x \geq 1
\end{cases}$ 
\(\displaystyle\lim_{x\to1^{-}} f\left(x\right) =\displaystyle\lim _{x\to 1^{+}} f\left(x\right) =f\left(1\right)\)
\(\therefore \:\: \displaystyle\lim _{x\to 1^{-}} x^{2} =\displaystyle\lim _{x \rightarrow1^{+}} 2-x = 2-1\) 
Hence, \(f(x)\) is continuous at \(x = 1\) 
Now, $ f'(x) = \begin{cases}
2x & \quad \text{for } x < 1\\
-1 & \quad \text{for } x \geq 1
\end{cases} $ 
\(\therefore \:\: \displaystyle\lim_{x\to1^{-}} f'\left(x\right) = \displaystyle\lim _{x\to 1^{-}} 2x =2\)
\(\displaystyle\lim _{x\to 1^{+}} f'\left(x\right) = \displaystyle\lim _{x\to 1^{-}} 2x = 2 \displaystyle\lim _{x\to 1^{+} } f'\left(x\right)= \displaystyle\lim _{x\to 1^{+}} \left(-x\right)=-1\)
\(\Rightarrow \displaystyle\lim _{x\to 1^{-}} f\left(x\right) \ne \displaystyle\lim _{x\to 1^{+} } f'\left(x\right)\)
\(i.e., L.H.D \neq R.H.D.\) 

Hence. \(f(x)\) is.no] differetiable at \(x = 1\)

A function's continuity specifies its properties and the value it serves as a function.If a curve is continuous at every point inside its domain and does not have any missing points or breaking points, the function is said to be continuous in nature.

If all three of the given conditions are met, then a function f(x) is said to be a continuous function at the point x = an in its domain. 

The three conditions are mentioned below:

The value of f(a) must be finite

If Lim _x→an f (x) is existing, is it essential that the right hand limit and left hand limit should be equal to each other, and both of them should be finite

Then Lim _x→an f (x) = f (a)

Thus, a function f(x) is said to be continuous at all positions in the range [a, b], including the endpoints a and b.

Continuity of f at a is: Llim _x→a+ f(x) = f (a)

Continuity of f at b is: Lim _x→b- f(x) = f (b)

In case of differentiability, if the derivative of the function f'(a) exists at every point within its defined domain, then the function f(x) is said to be differentiable at the point x = a.

The formula for differentiability is: f(a) =[f(a+h)−f(a)h]/h