静岡大 総合科学技術研究科 理学専攻 数学コース 数学 2024年第2回募集 院試 過去問 解答例・解説（全4問）

(1)

Let $g(x)=\alpha f(x)$. Take any point $a\in \mathbb{R}$ and any positive number $\varepsilon$. Since $f$ is continuous at $a$, there is a positive number $\delta$ such that $$|x-a|<\delta \quad\Longrightarrow\quad |f(x)-f(a)|<\frac{\varepsilon}{|\alpha|+1}.$$ Then $$|g(x)-g(a)| =|\alpha|\,|f(x)-f(a)| \le (|\alpha|+1)|f(x)-f(a)| <\varepsilon .$$ Therefore $g$ is continuous at $a$. Since $a$ was arbitrary, $\alpha f$ is continuous on $\mathbb{R}$.

(2)

Because $f$ is continuous, the inverse image of every open set of $Y$ is open in $X$. The set $$(V_1\cup V_2)\cap V_3$$ is open in $Y$, since it is obtained from open sets by taking a union and then an intersection. Hence its inverse image under $f$ is open in $X$. Equivalently, $$f^{-1}\bigl((V_1\cup V_2)\cap V_3\bigr) = \bigl(f^{-1}(V_1)\cup f^{-1}(V_2)\bigr)\cap f^{-1}(V_3)$$ is open in $X$.