Nonpiecewise Function Defined at a Point but Not Continuous There I make a big fuss that my calculus students provide a "continuity argument" to evaluate limits such as $\lim_{x \rightarrow 0} 2x + 1$, by which I mean they should tell me that $2x+1$ is a polynomial, polynomials are continuous on $(-\infty, \infty)$, and therefore $\lim_{x \rightarrow 0} 2x + 1 = 2 \cdot 0 + 1 = 1$.
All the examples they encounter where it is not correct to simply evaluate at $a$ when $x \rightarrow a$ fall into one of two categories:


*

*The function is not defined at $a$.

*The function is piecewise and expressly constructed to have a discontinuity at $a$.



I'd like to find a function $f$ with the following properties:
  
  
*
  
*$f(a)$ exists
  
*$f(a)$ is not (obviously) piecewise defined
  
*$f(x)$ is not continuous at $a$
  
*$f$ is reasonably familiar to a Calculus I student - trigonometry would be admissible, but power series would not (though they might
  still make for interesting reading)
  


There's always $[x]$, and $\chi_S(x)$ for any proper subset $S$ of $\mathbb{R}$.
