Question 37
a) Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that $g: \mathbb{R} \to \mathbb{R}$, $g(x) = f(x) + f(2x)$, and $h: \mathbb{R} \to \mathbb{R}$, $h(x) = f(x) + f(4x)$, are continuous functions. Prove that $f$ is also continuous.
b) Give an example of a discontinuous function $f: \mathbb{R} \to \mathbb{R}$, with the following property: there exists an interval $I \subset \mathbb{R}$, such that, for any $a$ in $I$, the function $g_a: \mathbb{R} \to \mathbb{R}$, $g_a(x) = f(x) + f(ax)$, is continuous.
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b) Give an example of a discontinuous function $f: \mathbb{R} \to \mathbb{R}$, with the following property: there exists an interval $I \subset \mathbb{R}$, such that, for any $a$ in $I$, the function $g_a: \mathbb{R} \to \mathbb{R}$, $g_a(x) = f(x) + f(ax)$, is continuous.
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