No, and in fact you shouldn't handle it that way. Template files are supposed to only contain variables and HTML. Instead you should place your custom function in a preprocessor and define a variable there. Then display this variable in the template.

Custom preprocessor (replace my_theme with your theme or custom module name):

function my_theme_preprocess_node(&$variables) {
  $variables['my_variable'] = 'value';
}

Twig:

{{ my_variable }}

Here is some reading on twig best practices: Twig best practices.

Some example code on preprocessors is here: template_preprocess_node(). Which is in fact systems invocation of hook_preprocess_HOOK().

Depending on the nature of your function, maybe but probably not.

As Clive pointed out, you can use certain functions in Twig. These functions however are mostly variable filters (formatters) for last minute alterations on variables. These can be used to translate, link, or in any way alter a provided variable. These functions are a combination of those provided by twig and some additions by Drupal. You can register your own Twig function, by extending on the TwigExtension class (example).

However most functions will probably be for adding additional data to the template and not to alter existing data. Instead you should place your custom function in a preprocessor and define a variable there. Then display this variable in the template.

Custom preprocessor (replace my_theme with your theme or custom module name):

function my_theme_preprocess_node(&$variables) {
  $variables['my_variable'] = 'value';
}

Twig:

{{ my_variable }}

Here is some reading on twig best practices: Twig best practices.

Some example code on preprocessors is here: template_preprocess_node(). Which is in fact systems invocation of hook_preprocess_HOOK().