5. (5 marks) Decide whether the following statements are true or false.
(a) If $f(a)f(b) < 0$ and $f(x) \neq 0$ for $x \in (a, b)$,
then $f$ is not continuous on $[a, b]$.
true
false
(b) If $\lim_{x \to 1^+} f(x) = 2 = \lim_{x \to 1^-} f(x)$, then $f$ is continuous at $x = 1$.
true
false
(c) If $f$ is differentiable at $x = a$ then $f$ is continuous at $x = a$.
true
false
(d) $\int_2^2 \sqrt{x^3 + 7} \, dx = 0$
true
false
(e) For all $f, g$: $\int f(x)g(x) \, dx = \left( \int f(x) \, dx \right) \left( \int g(x) \, dx \right)$
true
false
