Given topological spaces X,Y and a poin x_0 \in X, a function f:X-Y is said to be C_{x_0}-continuous if for every continuous curve w:[0,1]-X with  w(0)=x_0, the composition f*w is continuous at 0. We prove that, for every topological vector space X and every topological space Y a function f:X-Y is C_{x_0}-continuous if and only if f is sequentially continuous at x_0. We provide examples of topological vector spaces for which the C_{x_0}-continuity does not imply the continuity at x_0. We also prove a version of the intermediate value theorem for sequentially continuous functions f:X-Y. 

Engelking R. General topology. - M.: Mir, 1986. - 752 p.

Maslyuchenko V.K. Lectures on functional analysis. Part 1. Metric and normalized spaces. – Chernivtsi: ChNU Ruta, 2010. – 184p.

Baire R. Sur les fonctions de variables re´ elles // Ann. Mat. Pura Appl., ser.3. - 1899. - 3. - P. 1-123.

Young W., Young G. Discontinuous functions continuous with respect to every straight line//Quart. J. Pure Appl Math. - 1909. - 41. - P. 87-93.

Rosenthal A. On the continuity of functions of several variables // Math. Z. – 1955. – 63. – P. 31–38.

Ciesielski K., Glatzer T. Functions continuous on twice differentiable curves, discontinuous on large sets // Real Anal. Exch. – 2012. – 37, No. 2. – P. 353–361.

Ciesielski K., Glatzer T. Sets of Discontinuities of Linearly Continuous Functions // Real Anal. Exch. – 2013. – 38, No. 2. – P. 337–389.

Ciesielski K., Glatzer T. Sets of discontinuities for functions continuous on flats // Real Anal. Exch. – 2014. – 39, No. 1. – P. 117–138.

Aleksandrov P.S. Introduction to the theory of multiple general topology.–M.: Nauka, 1977.–368p.

Maslyuchenko V.K. The first types of topological vector spaces. – Chernivtsi: Ruta, 2002. – 72p.

Maslyuchenko V.K. Linear continuous operators. – Chernivtsi: Ruta, 2002. – 72p.