# Continuity of Functions

### Topics

**Intermediate Value Theorem (IVT):**

Let *f* be continuous on a closed interval [*a*,*b*]. Pick a *y*-value *M* with *f*(*a*)*b*) or *f*(*b*)*a*). *x*-value *c* with *a* < *c* < *b* and *f*(*c*) = *M*

*f* on a closed interval [*a*,*b*]:

*y*-axis where *f*(*a*) and *f*(*b*) are:

*M* strictly in between *f*(*a*) and *f*(*b**y* = *M*. The IVT guarantees that this dashed line will hit the graph of *f*. In other words, the IVT guarantees the existence of some value *c* strictly in between *a* and *b* where the function value is *M**f*(*a*) and ends at at *f*(*b*), then as *x* travels from *a* to *b* the function must hit every *y* value in between *f*(*a*) and *f*(*b*): *a,b*] skips a value, that function must be discontinuous: