The Intermediate Value Theorem

A continuous path must cross any given point between the end points of the path.

That's pretty useful information. Imagine a world where we couldn't count on this knowledge:

• Phrases like "I'll meet you half-way" would be meaningless. People could only reliably meet at destinations, unless those happened to be halfway to somewhere else.
  
• When travelling between two places, one might cease to exist at any given point along the route. The transportation industry would suffer.
• Crafty bank managers could claim that your money had disappeared when they tried to apply interest to your balance.

Clearly it would be chaos. Such a world would be a difficult place to live. A lot of the things that we take for granted would become much more tenuous, and thereby life would be much more randomly harsh.

Be happy to have the intermediate value theorem in your life.