If $f : \mathbb R^{n} \longrightarrow \mathbb R^{m}$ is differentiable at a point $c \in \mathbb R^{n}$ then it is continuous there.
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"Every linear mapping on a finite dimensional space is continuous"
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Proof that a linear transformation is continuous
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If $f'$ is differentiable at $a$ then $f'$ is continuous at $(a-\delta,a+\delta)$
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if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
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If $f$ is continuous then $G$ is homeomorphic to $X$.
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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 $\forall\alpha\in\mathbb{R}$
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continuity of $f(x,y)$ in $\mathbb R^2$.
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Differentiablility over closed intervals
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Continuous versus differentiable
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Let $f$ be continuous on $ [a,b]$, differentiable on $ (a,b) $ and $ f(x) \neq 0 \forall x \in (a,b) $ Prove $ \exists c \in (a,b) $ such that
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Is $p\mapsto \|f\|_p$ continuous?
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proof that $f'$ is continuous at $a$
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Proof of continuity implied by differentiability
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Continuity of a function in $\mathbb{R}$
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Proving that if $f: \mathbb{C} \to \mathbb{C} $ is a continuous function with $f^2, f^3$ analytic, then $f$ is also analytic
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If a function is continuous and differentiable everywhere is the derivative continuous?
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$f: \mathbb{R}^2-\{0\} \rightarrow \mathbb{R}$ is continuously differentiable and $f(\alpha x) = \alpha^2f(x)$, then $x \cdot \nabla f(x) = 2 f(x)$