Given the following statements about a function f: R → R, select the right option:

P: If f(x) is continuous at x = x₀, then it is also differentiable at x = x0.

Q: If f(x) is continuous at x = x0, then it may not be differentiable at x = x0.

R: If f(x) is differentiable at x = x0, then it is also continuous at x = x0.

Understanding Function Continuity and Differentiability

In calculus, understanding the properties of functions, such as continuity and differentiability, is fundamental. These concepts describe the behavior of a function at a specific point or over an interval. A function is said to be continuous at a point if its graph can be drawn without lifting the pen, meaning there are no breaks, jumps, or holes at that point. A function is differentiable at a point if it has a well-defined derivative at that point, which geometrically means the function has a unique tangent line at that point and its graph is "smooth" with no sharp corners or cusps.

Analyzing Statement P: Continuous Implies Differentiable?

Statement P says: "If f(x) is continuous at x = x0, then it is also differentiable at x = x0."

This statement is false.

Continuity is a necessary condition for differentiability, but it is not sufficient. This means that while a differentiable function must be continuous, a continuous function is not necessarily differentiable. A classic counterexample is the absolute value function.

- Example: Consider the function $f(x)=|x|$ at $x_0=0$.
- This function is continuous at $x=0$ because ${\lim }_{x\to 0}|x|=0=f(0)$.
- However, $f(x)=|x|$ is not differentiable at $x=0$. The graph of $|x|$ has a sharp corner (a cusp) at $x=0$.
- The left-hand derivative at $x=0$ is ${\lim }_{h\to 0^{-}}\frac{|0+h|-|0|}{h}={\lim }_{h\to 0^{-}}\frac{-h}{h}=-1$.
- The right-hand derivative at $x=0$ is ${\lim }_{h\to 0^{+}}\frac{|0+h|-|0|}{h}={\lim }_{h\to 0^{+}}\frac{h}{h}=1$.
- Since the left-hand derivative ($-1$) is not equal to the right-hand derivative ($1$), the derivative does not exist at $x=0$.

Therefore, Statement P is incorrect.

Analyzing Statement Q: Continuous May Not Be Differentiable

Statement Q says: "If f(x) is continuous at x = x0, then it may not be differentiable at x = x0."

This statement is true.

As discussed in the analysis of Statement P, there are indeed functions that are continuous at a point but not differentiable at that same point. The example of $f(x)=|x|$ at $x_0=0$ clearly demonstrates this possibility.

- The function $f(x)=|x|$ is continuous at $x=0$.
- But, it is not differentiable at $x=0$ due to the sharp corner.

This statement acknowledges that continuity does not guarantee differentiability, which is a correct understanding of the relationship between these two properties.

Therefore, Statement Q is correct.

Analyzing Statement R: Differentiable Implies Continuous

Statement R says: "If f(x) is differentiable at x = x0, then it is also continuous at x = x0."

This statement is true.

Differentiability at a point implies continuity at that point. If a function is differentiable at $x_0$, it means that the limit defining the derivative exists:

$$f^{\prime }(x_0)=\underset{h\to 0}{\lim }\frac{f(x_0+h)-f(x_0)}{h}$$

For this limit to exist and be finite, the numerator $f(x_0+h)-f(x_0)$ must approach $0$ as $h\to 0$. This implies:

$$\underset{h\to 0}{\lim }(f(x_0+h)-f(x_0))=0$$

Which simplifies to:

$$\underset{h\to 0}{\lim }f(x_0+h)=f(x_0)$$

This is precisely the definition of continuity at $x_0$.

Intuitively, if a function is "smooth" enough to have a well-defined tangent at every point (differentiable), it must not have any breaks or jumps (continuous).

Therefore, Statement R is correct.

Conclusion on Function Properties

Based on the detailed analysis of each statement:

- Statement P: If f(x) is continuous at x = x0, then it is also differentiable at x = x0. (False)
- Statement Q: If f(x) is continuous at x = x0, then it may not be differentiable at x = x0. (True)
- Statement R: If f(x) is differentiable at x = x0, then it is also continuous at x = x0. (True)

Thus, the combination of truth values is P is false, Q is true, R is true.