## Real Analysis Exchange

### Functions Continuous on Twice Differentiable Curves, Discontinuous on Large Sets

#### Abstract

We provide a simple construction of a function $F\colon\mathbb{R}^2\to\mathbb{R}$ discontinuous on a perfect set $P$, while having continuous restrictions $F\restriction C$ for all twice differentiable curves $C$. In particular, $F$ is separately continuous and linearly continuous. While it has been known that the projection $\pi[P]$ of any such set $P$ onto a straight line must be meager, our construction allows $\pi[P]$ to have arbitrarily large measure. In particular, $P$ can have arbitrarily large $1$-Hausdorff measure, which is the best possible result in this direction, since any such $P$ has Hausdorff dimension at most 1.

#### Article information

Source
Real Anal. Exchange, Volume 37, Number 2 (2011), 353-362.

Dates
First available in Project Euclid: 15 April 2013

https://projecteuclid.org/euclid.rae/1366030630

Mathematical Reviews number (MathSciNet)
MR3080597

Zentralblatt MATH identifier
1280.26021

#### Citation

Ciesielski, Krzysztof Chris; Glatzer, Timothy. Functions Continuous on Twice Differentiable Curves, Discontinuous on Large Sets. Real Anal. Exchange 37 (2011), no. 2, 353--362. https://projecteuclid.org/euclid.rae/1366030630

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