The problem asks us to determine whether two statements about differentiability and continuity are true or false. Statement (i): If $f$ is differentiable at $c$, then $f$ is both left and right continuous at $c$. Statement (ii): If $f$ is not continuous at $c$, then $f$ is not differentiable at $c$.

1. Problem Description

The problem asks us to determine whether two statements about differentiability and continuity are true or false.

Statement (i): If $f$ is differentiable at $c$, then $f$ is both left and right continuous at $c$.

Statement (ii): If $f$ is not continuous at $c$, then $f$ is not differentiable at $c$.

2. Solution Steps

(i) Statement: If $f$ is differentiable at $c$, then $f$ is both left and right continuous at $c$.

This statement is TRUE.

If a function $f$ is differentiable at $c$, then the limit

$$f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$$

exists.

If the limit exists, then the limit from the left and the limit from the right must both exist and be equal. Thus,

$$\lim_{x \to c} \frac{f(x) - f(c)}{x - c} = \lim_{x \to c^+} \frac{f(x) - f(c)}{x - c} = \lim_{x \to c^-} \frac{f(x) - f(c)}{x - c}$$

In order for the limit

$$\lim_{x \to c} \frac{f(x) - f(c)}{x - c}$$

to exist, it is necessary for

$$\lim_{x \to c} f(x) = f(c).$$

This implies that $f$ is continuous at $c$. Since $c$ is in the interior of $I$, we have left and right continuity.

(ii) Statement: If $f$ is not continuous at $c$, then $f$ is not differentiable at $c$.

This statement is TRUE.

This is the contrapositive of the statement: If $f$ is differentiable at $c$, then $f$ is continuous at $c$.

The contrapositive of a true statement is also true. We have already shown that if a function $f$ is differentiable at $c$, then it is continuous at $c$.

3. Final Answer

(i) TRUE

(ii) TRUE