Understanding Continuity of Functions

Question

I know that graphically a function $f(x)$ is said to be continuous in $[a,b]$ if there are no breaks in the curve for $f(x)$ in the interval $[a,b]$

I also know that by definition, a function $f(x)$ is said to be continuous at any given point $a$ if, we can find a $\delta>0$, for a given $\epsilon>0$ such that, the distance between $f(x)$ and $f(a)$ is less that $\epsilon$ whenever, the distance between $x$ and $a$ is less that $\delta$

But the problem is i am not able to "visualize" or "see" from the definition how it implies that the curve has no breaks. Some illustrations will be helpful

Thanks

Answer 1

A more helpful definition of continuity is the one given by Cauchy in 1821, in his work Cours d'Analyse. Cauchy said that a function $y=f(x)$ is continuous between two bounds if for all $x$ between those bounds, an infinitesimal $x$-increment necessarily produces an infinitesimal change in $y$; in other words, the value of $y$ can't "jump" and therefore there are no "breaks" in the curve.

A more detailed discussion of indivisibles and infinitesimals in the work of Fermat, Leibniz, Euler, and Cauchy may be found in this text.

Answer 2

Consider the step function, $$f(x) = \left\{ \begin{array}{ll} 0, & x \leq 0 \\ 1, & x > 0 \end{array}\right. $$ $f$ has a "break" at $x=0$. Observe that $f(0) = 0$. Now suppose $\epsilon = 1/2$ (really, any $\epsilon < 1$ is sufficient). Is there any possible value $\delta > 0$ that guarantees that if $|x - 0| < \delta$, then $|f(x) - 0| < 1/2$? What if the chosen $x$ is positive?