Functions — Question 9

Question 9 - Solution

(a) Continuity at $x = 1$:

Check: $$\lim_{x \to 1^-} f(x) = 2(1) + 5 = 7$$ $$\lim_{x \to 1^+} f(x) = 1^2 + 2 = 3$$

Since: $$\lim_{x \to 1^-} f(x) \neq \lim_{x \to 1^+} f(x),$$

The two-sided limit does not exist, so:

$$\boxed{f(x) \text{ is not continuous at } x = 1}$$

(b) Differentiability at $x = 1$:

A function cannot be differentiable at a point where it is not continuous.

So: $$\boxed{f(x) \text{ is not differentiable at } x = 1}$$

Explanation: The left and right-hand limits do not match at $x = 1$, so there is a jump discontinuity.