Recent questions tagged continuity

In this question, we only consider real-valued functions from $[0,1] \rightarrow \mathbb{R}$.Does there exist a function $f$ that is continuous at exactly $0$ and discont...

In this question, we only consider real-valued functions from $[0,1] \rightarrow \mathbb{R}$.Suppose $S$ is a finite subset of $[0,1]$. Does there exist a function that i...

Consider a bivariate function $f: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$.What does it mean to say $f$ is continuous at a point $(a, b) \in \mathbb{R} \times...

Consider a bivariate function $f: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$.If $f_{L}$ is continuous at the point $(a, b)$ for each line $L$ passing through $(...

Let $\mathbb{R}^{+}=\{x \in \mathbb{R}: x \geq 0\}$. For $x \in \mathbb{R}^{+}$, denote by $\operatorname{FRAC}(x)$ the fractional part of $x$, i.e., $x-n$ where $n$ is t...

Let $S$ be the set of functions $f:(0,1) \rightarrow \mathbb{R}$ with the property that there is a sequence $\left\{f_{n}\right\}_{n=1}^{\infty}$ of functions that conver...

For a real valued function $f:[0,1] \mapsto \mathbb{R}$, let $\omega_{f}:[0,1] \mapsto \mathbb{R}$ be the $oscillation$ of $f$ defined by\[\omega_{f}(t):=\sup _{|x-y| \le...

If $f, g:[0,1] \rightarrow \mathbb{R}$ are continuous and $f \cdot g=0$, then one of $f$ or $g$ is zero on an open subset of $[0,1]$.

Let $f: \mathbb{R} \rightarrow(0, \infty)$ be a function satisfying $f(x+y)=f(x) f(y)$ for all $x, y \in \mathbb{R}$.Prove that if $f$ is continuous at 0 , then $f$ is co...

Let $f, g:[0,1] \mapsto \mathbb{R}$ be monotonically increasing continuous functions. Show that\[\left(\int_{0}^{1} f(x) d x\right)\left(\int_{0}^{1} g(x) d x\right) \leq...