# Questions tagged [continuity]

Intuitively, a continuous function is one where small changes of input result in correspondingly small changes of output. Use this tag for questions involving this concept. As there are many mathematical formalizations of continuity, please also use an appropriate subject tag such as (real-analysis) or (general-topology)

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### Applying Fundamental Theorem of Calculus to improper integral

Let's consider $f:~]-\infty,\infty[~\to\mathbb{R}$ a continuous function. Our professor nonchalantly said that if we assume that $\int\limits_{-\infty}^xf(t)dt$ exists, then differentiating by using ...
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### $B\subset\mathbb{R}$ and $f:B\to\mathbb{R}$ is an increasing function. $f$ is continuous at every element of $B$ except for a countable subset of $B$.

I am reading "Measure, Integration & Real Analysis" by Sheldon Axler. The following exercise is Exercise 22 on p.39 in Exercises 2B in this book. Exercise 22 Suppose $B\subset\mathbb{R}$...
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### If a (partial) derivative is continous near x, does that imply that the (partial) derivative at x is either continous or non-existent?

Let $x \in \mathbb{R}^n$, and $A$ be a neighborhood of $x$. Let $f$ be a function with a (partial) derivative that exists and is continous on $A-\{x\}$. Does this imply that the (partial) derivative ...
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### What is the type of discontinuity of $e^{\frac{1}{x}}$ at zero?
The limits of this functions at zero are: $\lim_{x \to 0^+} e^{\frac{1}{x}} = \infty$ an infinity discontinuity $\lim_{x \to 0^-} e^{\frac{1}{x}} = 0$ a removable discontinuity The question is: Is ...