Showing that a function from a topology to $\mathbb{R}$ is continuous
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If $\{x:f(x) > a\}$ and $\{x:f(x) < a\}$ are open$\forall~a\in\mathbb{R}$ then f is continuous.
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Prob. 4 (a), Sec. 20 in Munkres' Topology, 2nd ed: Are these functions continuous in the product, uniform, and box topologies?
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Proving that this function is continuous on $G\times G$
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bijective continuous function on $\mathbb R^n$ not homeomorphism?
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Continuity of function mapping connected set to connected set
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A characterization of functions from $\mathbb R^n$ to $\mathbb R^m$ which are continuous
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Given a pair of continuous functions from a topological space to an ordered set, how to prove that this set is closed?
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Show that $f(x)=x^2$ is continuous at $a=2$ using the $\delta-\epsilon$ definition of continuity.
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How to show that the set of points of continuity is a $G_{\delta}$
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Proof verification: the product of two continuous functions is continuous
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$f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$
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$f(\mathbb{R}\setminus \mathbb{Q}) \subseteq \mathbb{Q}$ and $f(\mathbb{Q}) \subseteq \mathbb{R}\setminus \mathbb{Q}$ imply that $f$ is not continuous
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Show that exists a not decreasing function that $f:(a,b)\rightarrow \mathbb{R}$ that is continuous only in $(a,b)\setminus D$.
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Continuous function with compact domain has continuous inverse
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If a continuous function$f: (X, \mathscr T_X) \to (Y, \mathscr T_Y)$ is injective (Given $Y$ is Hausdorff), show that X is hausdorff
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proof $f(\overline{A})\subseteq \overline{f(A)} \Leftrightarrow f$ continuous
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$f:X\longrightarrow\mathbb{R}$ is continuous iff $\{x\in X:f(x)\geq\alpha\}$ and $\{x\in X:f(x)\leq\alpha\}$ are closed $\forall\alpha\in\mathbb{R}$